The name "Hammock Island" is a bit redundant. Nautical charts refer to the island as "The Hammock", which is a variation of "Hummock" which means a small island with trees in a marshy or shallow body of water. So, sorry, but it really doesn't have anything to do with the kind of hammock you swing in. But, we don't care. We like either interpretation.
We assume the island was once attached to the mainland, but the oldest charts we have found, dating to the early 1800s, show it as an island.