If you are thinking of renting (hiring) a car from an individual, JUST like at home, you would not be covered under the private passenger insurance that most individual owners have covering them. This is the same world wide, even in the UK. There is a stipulation in the insurance agreement that you will not be hiring out your car or the insurance becomes invalid. This is the reason that legal, legitimate car rental companies pay higher rates for their insurance coverage.
The only exception that you might be able to legally use would be if your husband, being Jamaican, borrows a car from one of his relatives while on the island. There would have to be a mutual understanding that no legal payment can be made for the use of the vehicle, or their coverage would not be in effect.
Please note that in Jamaica, most cars are insured for only third party coverage, meaning there is no coverage on any damages to the car itself, only the things it would happen to damage if at fault. Any damages incurred while driving the vehicle may have to be paid for directly from your own pocket.