I have what might be a better idea. Get a villa near San Gimignano or Siena. then day trip. You have a lot of moving around in a relatively close area. You can easily drive to Volterra, Siena, San Gimi, Monteriggioni, Certaldo, and a lot of other Tuscan towns from one location. We stayed near San Gimi and even went to 5 Terre and Pisa from there. We also trained into Florence from there, but we had been to Florence before, stayed in town, and saw the sites/sights. You will NOT need a car if you stay in Florence.
You could even do Assisi in a car from Tuscany. It is a little over 100 miles or so via good road (autostrade) between Florence and Assisi, certainly doable. You Arezzo and Cortona and Perugia on the way as well. Even closer from Siena.
Return the car anywhere you want. If you wander down to Rome to catch the high speed to Naples, you could stop by Orvieto and other places along the way.
If you go to Pompeii consider going to Herculaneum as well, much better preserved. Don't forget Ravello too, a great spot to overlook the Amalfi coast.