Evaluation: This tour can be a nice alternative to the walk through the interior of the island, passing through Kallítsos, the monastery of Taxiárches, Galaní and Panagiá and eventually returning to Chóra.  The path to Kallítsos is indeed one of the most beautiful trails in the Cyclades, but the circular walk requires you to follow a few long stretches of asphalt.
This walk back and forth on the contrary follows first the beautiful trail [1] until Kallítsos, but then continues with a stretch of path [10] and a nice gravel road to the monastery; after the visit, you can return to Chóra by a shorter route. Deserves ***.
[Update by Raymond on the 4^th of October 2015 and the 17^th of September 2017.]
[The translation of this walk was made with some help of Google translate - so, please, do not mind the mistakes...]

Estimated time: You need, in actual walking time (AWT), 45 minutes until the chapel of Páno Stavrós and then 40 minutes to reach Kallítsos or Kéndarchos. The way to the monastery costs 1h10. The way out via Kallítsos thus requires altogether 2h35 (AWT), but the return route will be shorter (2 hours). The actual time of this hike of 13,4 km thus amounts to 4h35, which means a long day of about 8-9 hours (in TWT). You should therefore take the bus Livádi - Chóra by about 10am. We walked from 10.30am to 6.30pm.
You can significantly shorten the hike by taking the bus to Kallítsos or until the monastery...

Route description: (0h00) You set off from Chóra and start this hike on the bus square of this village. In between the shop of Michail Bofilios and the kafeníon I Mili (The Mills) there is a flight of steps going down steeply. These steps bend to the left immediately and after 10 metres you go further down on a concrete path off the right. This path descends steeply again to the left after some 20 metres (red and white stripes =) and then finally turns into a paved street. A little further you will see a small red-and-white sign with the number 1 on a telephone pole – you will see these indications [1] or = regularly on this hike. Some 5 minutes later you leave the last houses behind you; the trail is now again covered with concrete. Another 20 metres further you DO NOT bend to the right - on a road leading to a cemetery and eventually back to Livádi…. Instead you take the stony little road going straight, in the direction of a high aerial (hiking sign). When looking in the distance you will be able to discern the trail going over the next slope and going up to a chapel with a white arch – you will reach that spot later on! The trail descends away from the road, to the right, next to a stable (=). You descend now to a stone bridge, painted in white. To the left of this bridge there is a watering place with a source – [1] on the balustrade of the bridge.

(0h12) The trail goes up again now (=), sometimes it even turns into a nice flight of stone steps (= and [1]). Almost on top, at the foundations of a stone building, there is a path to the right (to Livádi or Psili Ammos, see the hike Chóra - Psíli Ammos). Your trail, however, goes straight [1], in the direction of the chapel with a small arch. The views on Chóra behind you and on the bay of Livádi get the more and more beautiful.

(0h21) Ten minutes after the watering place you get to the top, near a small farm and the chapel. You leave a gravel road on the left and the trail descends straight (hiking pole [1]). You descend now  to a second valley with a bridge with an ugly metal balustrade. You reach this bridge after another 5 minutes – the valley is fairly green, even in Summer.

(0h26) Your next objective now is the chapel with the little white arch, which you see halfway to the top of the big hill on front of you, a hill with lots of aerials on top. The trail clearly continues to the left [1], in between walls: the sunken path is sometimes blocked by vegetation, but after almost 5 minutes, you get to more open ground, with a little valley in the middle.

(0h30) Attention: there are two paths here and you should take the left path, to the left of the valley (red arrow + [1]). 
A little further, you have to go up on the left on rocky ground,  in the direction of two converging walls – still in the direction of those aerials.
Now, you walk during 7 minutes between stone walls, then the chapel appears again: you go always straight in that direction, you descend and go up again, always straight ahead towards the chapel with the white arch.
Just before you reach the chapel, you get to a recent gravel road that comes from the right: you follow it to the left and then you zigzag towards the chapel.

The chapel of Páno Stavrós, between Chóra and Kallítsos.

Above you have a wonderful view on Chóra, Livádi, the beach of Agios Ioánnis and the island of Sífnos. Therefore, this is the ideal spot to take a break; the chapel is open, but also the steps in the shade are very inviting to sit for a while!

(0h44) At this chapel a beautiful kalderími starts off (=) and you will follow this one until Kallítsos. You go up gradually, in a rocky and very impressive landscape with great views on Agios Ioánnis, Psili Ammos, Livádi, the little island of Vous, Sífnos and Chóra behind you.

The kalderími between Chóra and Kallítsos.

Often, there are nice curbstones, sometimes a nice pavement, and especially after some 18 minutes, there is a stretch with a beautiful pavement. You might even wonder who has ever put all this work into this wonderful trail. After a climb of about 23 minutes you reach a kind of pass on a protruding cape [1]; this is Vounáki - you can now see Kallítsos and also the large road above it.

[If you prefer to follow a shorter route to the monastery, without going through Kallítsos, you could take the small road that goes up here to the left, to the gravel road that you will follow during your return route. You could, after 4 minutes, follow this road to the right - and so you could reach the monastery after a little less than 40 minutes, while the route that continues straight to Kallítsos will cost you 1h30…]

(1h07) You descend to Kallítsos for about 10 minutes – after 10 minutes, there is a triangular spot, where according to the map a path mounts to the left. It finally gets to the gravel road you will follow later on (see under point 2h08), but in October 2015 it was too overgrown.
You thus take a right and finally, the long, beautiful path reaches the asphalt road that was constructed in 2003. Some 100 metres in front of you, you can notice the first houses (hiking pole).

(1h21) You follow the road for a short while to the left and when you get to the sign Kallítsos, you go down the brick staircase on the right (2 red dots + [1]) – this staircase gets down to the old trajectory of hike [1]. You enter the village by taking a sharp left and you continue going down the staircase towards the ‘centre’; you get to a cross street where you take a left, and a little further, you keep to the left again: so you will finally get to a kind of square. Over there you find a tap where you can freshen up and a wall where you can eat your sandwiches in the shadow – unfortunately there is no tavern or café. But still, this is a lovely spot, underneath a palm tree and a pine, isolated and peaceful.

The "platía" in Kallítsos.

(1h27) After this break, you continue straight, then you take the nice staircase that rises to the left; you veer to the right and you reach a wide paved street that leaves the village while going up. The street becomes a concrete road and also beyond the very last house, you continue to go up on concrete. You go up to some ruins where you veer left and so you get to the actual route of walk [10] (hiking pole).

(1h36) Now you go up to the left on a concrete road (signs). You cross the asphalt road after 1 minute and continue straight across (red sign Profítis Ilías 35 minutes).

The hiking signs and the information panel for walk [10].

You pass to the right of some ruins and get to a very rocky climb – at first to the left of a ruin, then for quite some time between walls. You have a beautiful view of Kallítsos and you arrive after about 8 minutes at the top, next to a large round ruin.

(1h46) You veer slightly to the left (=) between the walls and you still go up a bit until you get to a slight depression between your hill and the next one - on the right, you enjoy a nice panorama on Kýthnos and on the islets of Vous and Pipéri. You continue flatly for some moments, but at the end of the depression, you do NOT go up to the right between walls, but you keep to the left (=) in order to follow a clear path towards a grove. After an easy end, you go up again, always with a beautiful view of Kallítsos and the bay of Kéndarchos. About 5-6 minutes later you descend slightly to the left (=) and you reach an easy path that later on continues next to ruins until a cross valley with a lot of vegetation.

(2h00) The small sign [10] points obliquely to the right and you go up while winding between ruins and rocks (=), on a rocky slope or on rocky steps. You thus arrive after 4 minutes between a wall and a nice grove of oak trees and so you get to a gravel road (red sign [10]).

(2h08) You take a right and soon you veer to the right – with  Galaní to the left and in front of you the chapel of Profítis Ilías – and then left again (sign [10] and =). You pass beneath a rather elegant building with a dovecot and you finally arrive near the small nice chapel of Profítis Ilías - it is closed.

[Walk [10] continues to the right here – see the hike Kallítsos - Profítis Ilías - Triantáfyllos - Panagía Skopianí - Kallítsos...]

The chapel of Profítis Ilías and the hiking signs for walk [10].

(2h16) You now already can see, on the left, the big monastery and you follow the gravel road in this direction. You descend slowly, always with a beautiful view of the valley, with Pýrgos on the left and more to the right the village of Panagiá with a tip of the red roof of the church. After 8 minutes, you see on your left some remains of the marvellous monopáti that was here before the construction of the road... A little later, you pass to the right of the chapel of Agii Apóstoli - you veer to the right, straight to the monastery, and the gravel becomes concrete. While having a beautiful view of the large building, you go down to the asphalt road. You reach this road a little beyond the monastery, and so you should turn right onto the road for a few more minutes... (2h35)

The impressive monastery of Taxiárchoon

This monastery was built like a fortress, it is situated some 100 meters above the sea and it dates from the 16th century. You can enter the monastery through a very low door – if there are not too many tourists the only remaining monk will offer you some loukoúm and he will show you the church with its beautiful ikonostási, originating from 16th century Russia. But often the door remains closed...

(2h35) You start your way back and you return first on the road for 3 minutes: in the bend of the road (you already see Galaní on the right), you take the concrete road that rises sharply to the left.

[If you want to continue via Galaní and Panagiá in order to return to Chóra, you can follow the second part of the hike Chóra - Stavrós - Kallítsos - Moní Taxiárchoon - Galaní - Panagiá - Gýftika - Chóra. This route will take approximately 2h20, while the return by the same route will last for about 2 hours...]

The concrete road rises quickly and passes next to Agii Apóstoli after 10 minutes; after another 12 minutes, you arrive near Profítis Ilías – you again will follow the itinerary of walk [10].

(3h00) You veer right, you pass the pigeon-house and after another 9 minutes and just beyond the oak grove, the trail [10] goes down to the left.
You will now continue to follow the gravel road, in order to follow a much shorter route - the walk to Vounaki will take only 12 minutes, while the road following the same route as during the way out would take 50 minutes!

(3h09) Near the sign [10], you thus continue straight on: the dirt road is quite pleasant and follows the slopes of the Diaséla, above Kallítsos and the path that you just have followed. After 8 minutes, a small road goes left, in a bend and next to a pole.

(3h17) So you go down LEFT, but in the first bend of the road, you continue straight on a path, which now descends very quickly on the crest of the hill, with Kallítsos to the left of you.

(3h21) Already after 4 minutes, you get to the main path [1], on cape Vounáki: you take of course a right here (hiking pole). The continuation of the walk on the beautiful kalderími is very clear; especially after 3 minutes, there is a very nice stretch, with a beautiful view on the beaches and on the bay of Livádi and Chóra. Finally, you descend to the right of the chapel of Páno Stavrós.

(3h43) You zigzag down beyond the chapel and follow the road, but beyond the bend, you can continue on the path to the right (red sign Chóra 50’). A rocky trail descends first to the right of a wall, then it continues meandering between walls (= =) - toward the end, it is quite narrow because of the vegetation. You thus arrive on an open slope, where you continue your descent straight following a stony path.

(3h58) You arrive in a little valley and – pay attention – you should keep to the right here, between walls (red arrow on a stone). The following stretch is quite difficult because the sunken path between walls is overgrown ) and halfway, there is an obstruction.

(4h03) You reach the bridge with metal railings and you cross it to the right. You follow a nice path until you arrive next to a pigeon-house, then you climb again towards the chapel and the small farm with a lot of rubbish - there is a small road that arrives from the right.

(4h09) You continue straight ahead (red signs), to the right of a wall, and you go down for 8 minutes on a wide path, until you get to the second bridge [1], with to the right the source which lies in a very green valley.

(4h17) You go up again for 7 minutes and continue straight on the little road; after 1 minute, you get to the bend of the road that descends to the left towards the cemetery. You continue to go up straight on the concrete road; it becomes paved, veers to the left, goes up again and turns right - and you finally take the staircase up to the left until you get to the platía. (4h33)