Maybe not. If the car you have is corporate car and attempt this at a franchise location they may not allow it.

Very true. Someone else posted here they did a paid one-way from MSP to a franchisee in Missoula MT. They switched from the expensive one-way rate to a roundtrip local rate, but also had to switch into a locally owned car.

However, both locations your rental involves are likely corporate, since they both accept the FL driveout. So they probably will not care, or might even be happy they don't have to clean the car and can immediately re-rent it out.