You can hang at designated areas in FL state parks now, you just have to use tree straps. The great thing about the primitive are at Torreya is it is high on a bluff overlooking the Appalachicola river and there is some good hiking trails there.
You mentioned the Florida Trail. The Apalachee Chapter in Tally will be doing a hike of the 150 miles that they maintain in November. I plan to join for at least 10 days and will be taking the hammock. Both the Panhandle and Apalachee Chapters have meetups with all their hikes listed.