American Eagle. No seriously... especially from SJU. Otherwise you'll have to look for small airlines that fly between islands. For example, SkyKing is a Turks and Caicos airline I frequently used, but they only fly between the islands in the Turks & Caicos, Haiti and DR. So not much help in your situation. I am sure there are little puddle jumper airlines that can help from islands like SKB.
They fly Dash 8s on a variety of routes around the Caribbean - it operates almost like a bus service, in that many of their routes involve multiple stops on circuitous route between a number of islands.
I flew CS from Barbados via St Lucia to Antigua and then also a return for the day from Antigua to St Kitts.
I posted a trip report on my trip in the Trip Reports section.
From those islands you have mentioned your optioins are really limited to Caribbean Sun (as it is based in SJU) and American Eagle (alse based in SJU). LIAT and Eagle are the only two that can take you to SDQ though. In those islands you mentioned Caribbean Star only flies to SKB.