Only if you are within range of a relay tower which will only be when very near the ports of Key West or New Orleans. I am not sure if your phone will be compatable with any that may be in Mexico, Grand Cayman, or Jamaica but it is very doubtful and even if they were the cost would be astronomical to say the least. Now if you have one of those Satellite types then it may well work but few folks actually have those.
This is one of the most complicated questions ever... the short answer is "no" but there are ways you can get connected. The thing to do is to contact Cable & Wireless to see if you can set up a temporary account. (your local service does not provide coverage in the Caribbean - even with roaming charges).
There are other avenues as well, I am told. Call your cell-phone provider and see if they can provide international coverage for you in areas compatible with their system.
Either way - it is complicated and expensive. I recommended doing what I do - visit the local Internet cafe in each port and send email, but first get yourself an account with HotMail!