Umbria/Tuscany

Is Perugia the closest place to rent a car?
Perugia Airport is the closest to Assisi.
Foligno is also not far (Europcar has an office there).
How do you plan to get to that area and from where? If coming to Assisi by train from Rome, Foligno or Spoleto, along the rail line, have offices.

towns in-between Assisi and Florence:
The sky is the limit here. Some popular destinations are:
Cortona, Montepulciano, Pienza & Monticchiello, Siena (a small city rather than a town), Monteriggioni, San Gimignano, Chianti Hills, just to name a few.

Since you only have the car for three nights, I would suggest making your main base in Montepulciano. The area of Tuscany south of Siena is the most beautiful part, IMO. I would spend two nights in Montepulciano, perhaps you could swing by Cortona on the way in. Then spend the second day driving around the area between Montepulciano and Montalcino. RS' Florence and Tuscany book has good suggestions for drives to connect all the towns. The third night I would move to Siena, even though it's only for one night. It's a good contrast to all the small towns in Tuscany. Then when you're leaving Siena on the 21st, you could visit San Gimignano or Volterra on the way to Florence to drop off your car. It's not really enough time to see all that Tuscany has to offer, but it's probably the best way to make use of the time you have planned.

Note that I am not trying to get you to change your plans. But we chose to stay at a remote (zero independent restaurants nearby) countryside luxury hotel in Sinalunga (Locanda dell'Amorosa) for the towns we were targeting. This is related to our using our lunch as the main meal, and only having bar snacks, at the hotel, in the evening. (Aging digestive systems, let's say.) We also went to "destination" winery restaurants almost every day.

Anyway, we have no interest in Agriturismos, but liked the effortless departure, return, and parking at a remote, small hotel. Having (for example) to lock a stanchion across our reserved parking space is drag, sometimes. Pool and frigid air conditioning useful in summer, believe me.

I would note that you're not far from Castiglione del Lago. We didn't sleep there, but enjoyed a low-key visit. But I've read that it has some nice restaurants. And it's a bit off the beaten track. OTOH, Cortona is a lovely town, but overrun in the daytime, and a steep hill drive home. Public parking difficult in the daytime. If you stay in this area, consider our favorite winery and restaurant of the trip, Avignonesi . (Expensive.) Advance reservation and payment required. We had our hotel do it for us, long before arrival.

Thank you for all your recommendations. To answer the question above, we will be traveling to Assisi from Rome via train. We don't need a car for the time we are in Assisi and plan to pick up the car once we leave Assisi which makes me think that Perugia is the best option to rent the car as it is in the general direction we are going.

In Assisi you look across the valley at Spello, a beautiful , flower filled town where we spent two weeks. I highly recommend that you stay in Spello. It’s easy to take day trips to towns such as Montefalco and Bevagna.