French Florida: Surprisingly hard to do
Spain was very sensitive to the possibility of other European powers colonizing Florida because their treasure fleets passed by there on their way to Spain, and after the abortive Huguenot settlement they spent a great deal of money and effort colonizing Florida and setting up Indian missions. Your best bets might be the initial Huguenot colony (though it was too close to Spanish centers of power in the West Indies for success to be too likely) or a successful revolt of the Florida Indians. There was a revolt in the first half of the 1600s--probably in the 1630s if I recall correctly. The Timucua and to a lesser extent the Appalachees were involved. Historically the Indian revolt was kind of a half-hearted affair.
By the time the Spanish were weak enough in the West Indies to have allowed a French colony in Florida they were strong enough in Florida that it wasn't really worth challenging them there. Plus the French were largely sidelined from Great Power games by their wars of religion for much of the last half of the 1500s.
Historically the British smashed Spanish power in Florida in the early 1700s by crushing and enslaving the Apalachee Indians, who had been the major Spanish Indian military allies in Florida, then carrying off most of the mission Indians as slaves. That left Florida without a labor force and unable to support much of a Spanish garrison without help from the outside. So at that point Florida was open for the taking, but Spain and France were allies then--both ruled by Bourbon kings.
Probably your best bet for France to end up with Florida would be for the Huguenots to actually found a colony in what is now South Carolina, further from Spanish power in the West Indies and less threatening to it initially. Spain would probably still attack, and probably still try to found a colony in Florida, but they would be further from their bases when they attacked and if the French held on the Spanish might settle for occupying and defending Florida, at least in the short term. If there was still a French colony in South Carolina when Spanish power in the West Indies started to collapse in the 1600s, then the French might well extend their power into Florida. I'm not sure that's what you are after,