There are variables for that decision. Will you be staying in Rome or going straight from the airport? If you'll be in Rome, driving in the city is a concern, but picking it up in the Villa Borghese parking lot, listed with the via del Galoppatoio address, makes for a drive that isn't too bad. If you pick it up at the airport, the driving is easier but there's a 14% surcharge on the whole rental.
Another complication is that the Avis office in Chiusi, which is the Auto Europe affiliate closest to Montepulciano, is reportedly closing down. You can still get a car there arranged through the Siena office for a surcharge of around 100 euros. The same thing is also happening in Orvieto, which would be another alternate place, but there's a Hertz office there, so you could get a car if arranged specially through Auto Europe or Hertz directly. So it's your decision; it depends how comfortable you are driving in and around Rome.