Many Tarpon migrate north to south in cooler months and south to north in warmer. Some of them do the offshore to beach/inshore circuit as well, especially around spawn. You are always going to find local fish in south florida and the keys that don't move around as much, but they are a bit more finicky eaters than the ones making the migrations.

I've also found they tend to feed at night all the way to mid morning more, with big bursts of activity around bait pods happening from first light to an hour or so after sunrise. Around the pods, match the hatch is the safest bet, keeping your live bait on the edge of the pods or just below them so they look like stragglers. You can also do pretty good by using some of the other predators feeding on the bait, like lady fish, lizard fish, etc. or of you know threadfins are flashing around the area, you can never go wrong sabiki-ing a few of those up to free-line. Night time on out going tide you can do pretty good with crabs and even shrimp. They will sit around the passes letting meal come to them.

When it is cold, murky like during the winter months, you can do pretty good down south sinking mullet, whiting, ladyfish on the bottom ...preferably live, but dead works too if you can avoid catfish and the like. Providing of course you know they are around. If it has been cold for a few days, the water dark or chocolate brown from a good blow, and there is no bait around most of the Tarpon will have moved offshore.

Just my .02