Looks like the (Long awaited) weekly connection from Skiathos has materialised, The Skyros shipping company ferry 'Achilleas' will operate as follows ...
Sunday ferry service to Skyros
Volos 06:45 Skiathos 09:25
Skiathos 10:00 Skopelos 11:05
Skopelos 11:20 Alonissos 11:55
Alonissos 12:05 Skyros 14:45
The service from Skyros is on a Saturday
Skyros 12:15 Kymi 14:00
Kymi 15:00 Alonissos 17:20
Alonissos 17:35 Skopelos 18:10
Skopelos 18:40 Skiathos 19:40
Skiathos 20:15 Volos 22:45
It is only mid-June (20th) - Mid September (11th), so you have to choose that period
The ferry also goes 2/3 times a week from Skopelos/Alonissos around 12:30 arrives Skyros about 20:00 with a connecting bus to Chora. return is at 07:00.
It is only mid-June (20th) - Mid September (15th), so you have to choose that period otherwise there is no easy way except from Evia.
The timetable is here http://www.sne.gr/