There are actually at least five airlines that do inter-island flight, but not all of them do every station:

Hawaiian
IslandAir
Pacific Wings
Mokulele (branded as go! Express)
go!

If at all possible, please don't fly go!

Their business practices (evidenced by a multi-million dollar judgment against them) helped drive Aloha Airlines out of business, thus hurting a large number of local families.

They also have the single most uncomfortable aircraft I've ever been in, the CRJ200

Any of the four airlines would do what you need to do, but each has its idiosyncracies