The other option over buying/borrowing a phone there is to unlock your current device and use a Jamaican SIM card in it. This would work with just about any phone from AT+T or T-Mobile (or any other GSM provider). Will not work with CDMA (Sprint/Verizon), but they do have a few dual devices (Like iPhone 5) that have a SIM slot to use GSM service.

I know T-Mobile will allow you to unlock your phone after 90 days of service, you call customer care, let them know you'll be travelling, and in a few days they send you an email with the unlocking instructions. AT+T is not so easy about it, I think you may need to be out of contract to get them to do it.

Once you have an unlocked device, you can put a SIM card in it from just about anywhere in the world, no need to buy a phone, just the SIM and minutes.