The newly revamped, post-hurricane Little Dix Bay in the BVI looks rather special. I knew it in the 1990s and the beach setting there is hard to beat. Ditto old-timer Curtain Bluff on Antigua and those two make a nice combo. June is a good month for the Caribbean - lower prices and gorgeous climate.
Unfortunately, most hotels in French Polynesia are chains - FS, IC, Meridien etc - but Le Bora Bora is locally owned, recently refurbed, and has the best views of the mountain and lagoon.