Strangely enough, a Goodwill or other thrift store is a perfect place to find winter stuff in a warm climate. Why? Because people buy the warm clothes for that one-time trip, or move from a cold climate, and most of their stuff goes to a second-hand store. Since moving to Arizona, I have pared down my massive sweater collection and still have a few coats that should probably be donated too.
Or wait until after Christmas and buy things when they go on clearance. They might not have as much in FL so it's a gamble.
Also, when my son needed a parka for an overnight up to the mountains, I borrowed one from a friend who had just moved here from a cold climate. You might have to buy some things, but perhaps you could borrow some too if it's a one-time trip.
PHXscuba