10 days is far too short to do Athens and 3 islands, you lose a day on each ferry journey. A ferry to Santorini 7 hours, 4 to Paros for example.

It's backpacking in the islands generally, flexibility is the key, particularly if you wish to reach the less touristy ones.

Most seasoned Greek island hoppers never book accommodation in advance and travel out of hi-season when accommodation is never full. Room and hotel owners are very flexible about the length of stay and only require 1 day notice of when you are leaving or if you want to stay a few days longer, no problem.

Accommodation is easy to obtain on arrival at any island, except for mid July/August, the best price is usually obtained by bargaining direct with the hotel/rooms for rent place.

Island hopping is easiest when you go to islands that are on regular year round ferry routes (i.e. predictable). e.g.
Athens/Piraeus-Paros-Naxos-Ios-Santorini.

It is easy to fly to somewhere like Rhodes, and island hop up the Dodecanese islands.

It does take a few trips around the islands to 'get in tune' with the complex Greek ferry system.

It is far cheaper to take ferries as a foot passenger and then rent a car if required on each island, I don't think hire car companies allow you to take them on the ferries at all.

A useful guide book is 'Greek Island Hopping' pub: Thomas Cook.