It's not exactly clear if you'd already planned for this but you definitely don't want a car for Rome either (as you've already been counseled in a previous thread). Also, you've only planned 3 nights/2 full days for that city; it could use another one.
At a glance, I'd be strongly tempted to skip the anything south of Rome; you really don't have adequate time to cover Spello>Bari>Amalfi Coast, and as already pointed out, you don't want a car in Positano. Fly into Florence, do your sightseeing there, then rent a car and work an itinerary for the Tuscan/Umbrian towns/countryside between Florence and Rome. End up in Rome, dump the car, spend 4-5 nights there and fly home from FCO versus backtracking to Florence.
You could also extend your time in Florence to include some day trips, or move on to Siena by rail or bus, spend 2 nights there, and then rent the car. One note of caution? It's not a good idea to leave anything at all you don't want to lose in an unattended rental vehicle.