these days there aren't much in the way of cell phone rentals......use to be quite a few some years back....
either bring a unlocked phone or pick one up in Jamaica......lots of cell phone stores here so just find one....
not a good idea to start using your North American phone in Jamaica....you risk coming home to a huge cell phone bill.....lol
of course if you know what you doing.......no problem
guess a lot of vistors these days just use the wifi on der North American phones and use WhatsApp or FB messenger etc to communicate......

Cool Runnings, Marko