Similar places to Little Corn Island

Hi, I visited Little Corn Island with my girlfriend this May and we had a great time. I imagine it's the closest you can get to how the Caribbean was a hundred years ago. Does anyone know of any other similar places in the world? I am in the early stages of planning a trip in February and I would like to go someplace different but similar. The following is my criteria:

-You can have a beach almost entirely to yourself
-Relatively undeveloped tourism
-Cheap (we stayed for $30 a night and most meals were fruit off the trees)
-Good vibe, place has a loose conception of time

Edited by: s2000ftw