Just had lunch with a friend who used to live in the Caymans. They used to sail down to Swan Island to go diving
. He used to do this late in the season and into hurricane
season because the trades tend to shift a bit SE that time of year.
So, Rio-Cayamans-Jamaica route
based on a mix of my local knowledge and his: The winds tend to be light along the Honduran coast (south the Bay Islands) due to the land effect of Honduras
. Once you get up to the latitude of the Bay Islands you get into the Trades...often really pumping just N of Roatan. For the least time pounding to windward, you could run along the Honduran Coast, then up to the Bay Islands, then Bay Islands to Caymans (optionally stopping in at Swan). From there, no matter what time of year, going east in prevailing conditions is going to be a bitch. You could wait in the Cayamans for a front and then catch that to Jamaica or just pound it out to Jamaica. This is why people don't tend to go this way.
Personally, if I were to leave the Rio for the Eastern Carib, I would probably take the up-and-over route
via Mexico. Getting through the straits of Florida
can present it's challenges due to prevailing winds, but you can work you way through the Keys or wait for a front. Down through the Eastern Carib and then downwind trades sail all the way back to the Rio.