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Why not rent a car in Montepulciano? It is about an hour from Assisi. Then drop it off at the Rome airport. I have used both Autoeurope and Gemut.com, but have not rented recently. The selection of cars may be less in Montepulciano, but it is a lot of travel to get to a bigger centre.

If you decide to drop your car at the airport, I would advise you not to drive in Rome. Go directly to the airport with the car. If you wish to see Rome, drop your car at the edge of Rome. The use transit to navigate in Rome.

Have a great trip.

There are no car rental agencies in Montepulciano
Closest would be Chiusi-only Avis/Budget

Your best selection will be in Florence at train station or Perugia

Why not pick up and drop off the rental in Florence? You could then take the train from Florence to Rome. Only reason for this suggestion is to avoid driving in Rome, or feeling “stuck” at FCO after dropping off the rental car.

for more context; we will have already visited Rome at the beginning of the pilgrimage so I thought we would just drop it and go to our Rome airport hotel. I we rent to/ from Florence train station how would we get to the FCO hotel? Train to Termini and then cab to airport hotel?

I’d just drop at FCO in that case

Given the additional info, I agree with ChristineH just drop at FCO.

I would rent in Perugia and return it there, then catch the train to Rome. Alternatively, if you rent from Avis-Budget in Perugia check if it is possible / affordable to drop the car in Chiusi instead. Chiusi is closer to Montepulciano and trains to Rome take less time from there vs. from Perugia. The Chiusi Avis-Budget office is closed midday for several hours as a heads up.

Unless you want to go to Florence I don’t think it is worth going all the way there just to get a rental car.

I don’t particularly enjoy driving so when I can drop the car and hop on a train that is my preference.

^^^ this. Do this ^^^

From Assisi I would rent in Perugia (rental offices are located both at Perugia city center or Perugia airport, which is located between Perugia and Assisi).
After visiting Montepulciano area, if you are going to Rome FCO, I would drive to FCO and return it there. Alternatively you could return it in Chiusi and take the train from Chiusi, but Avis/Budget is the only choice in Chiusi.
Driving from Montepulciano to FCO airport is an easy 2.5 hour drive (100% freeway once you enter the A1 freeway at Chiusi-Chianciano exit).
You can also return it back to Perugia, but it's a major detour. From Montepulciano to Perugia you need well over one hour. You can be half way to Rome in that length of time.
Just avoid driving through the Rome's G.R.A. (Grande Raccordo Anulare, or Grand Ring Beltway) during commute time on a weekday, as there would be traffic, but otherwise it's a breeze outside of rush hour.
Before reaching the airport, you need to refuel your rental car. This is where I usually stop for refueling. It's the last Freeway Service Rest Area before heading to the airport, without needing to exit the freeway for gas. Save it in your phone.
https://goo.gl/maps/emp3ZYbYJNC7XRoe9

Another option is Arezzo. We are renting in Florence as we travel south and returning in Arezzo & it wasn’t much more to do so. We’ll be in Montepulciano and didn’t want to go back to Florence.