I didn't have a JA cell phone on my first trip. In my experience, hotels and restaurants will let you use their phones if you give them the number to dial. (I guess they just want to make sure that it is a local number and you are not calling overseas.) Or, if you are happy with a driver and want him to pick you up after dinner, etc. he may go to the restaurant manager, give them his number and have the restaurant call him when you are finished with your meal. Doesn't get more convenient than that. You'll find that most Jamaicans in the tourism industry are very accommodating as they want to keep your business.
But yeah, a couple times I did suck it up and call a driver from my US cell phone (with an international plan), when I was stuck somewhere without a taxi nearby. I have since acquires a Jamaican cell phone now that we visit the island on a regular basis.