Rental Car in Siena?

Staying 2 nights in Siena take the bus there and then rent the car when you leave there.

The only reason to rent the car in Florence would be if you wanted to drive the SR222 Chiantigiana scenic road down to Siena. It's a beautiful drive, but given your schedule constraints I doubt it would be the best use of your time.
Otherwise I agree with Jazz that your better option is to take the train to Siena, find accommodation in or near the center of the city to make exploring a bit easier, and then rent the vehicle as you're leaving.
There's a Hertz location located about 2 blocks from the train station that's quite handy, though it doesn't open until 9 AM. If you want an earlier start on getaway day you could collect the car the afternoon before and park at the cheap underground lot under the train station - costs 2 Euro per day. We never had any issues finding a spot during a visit earlier this year, but others have reported problems finding parking during festivals or special events, so double check what's going on in the city if you opt to do it that way.
Having a car is a great way to explore the countryside and hill towns in Tuscany and Umbria.

Your intent for Siena isn't really clear. If you only plan to visit sights in Siena for 2 days before moving on to Perugia/Assisi, it would be better to stay in town and delay picking up a car. But if you plan to take day trips from Siena on both of those 2 days, then it might make sense to pick up a car in Florence and stay in the outskirts of Siena.

P.S. I prefer the direct bus to the train, but either works.

I think you’d want to see the towns outside of Siena too and drive the Chiantigianna, SR 222. We have stayed in several of those charming towns. I would have a car while in Siena. There are parking lots at top of the hill.

We visited Siena from another town by car in July, and parking was very difficult to find. The city was a long slog "uphill both ways (!)" Because Siena is an especially rich, urban (not just "hill town") destination, I would not pick up the car until you leave. I wouldn't want to have to wait for a bus into and out-of Siena for two days of visiting.

There is no question that a car is useful for general Tuscany purposes, but it is a hindrance for Florence or Siena visits. (Don't forget to read up on ZTL.)

I would check whether the Siena cathedral floor is uncovered any days that you could choose to be there. Ticketing visits in Siena can be horribly time-consuming as a walk-up. I would pre-buy just avoid the lines, even if I weren't confident of getting "full value" from a multiple-attraction package. Find the official seller, not some tour company.

There are also two parking lots with free escalators to the top of the hill (which we have used recently) and the map says there is a third one. However, I'm in the camp with those who recommend train or bus to Siena, stay in and see Siena, and then rent your car. We rent cars all the time in Italy when we need to, and it's fine, but in this case you don't need to.

It might depend on what type of car you need- manual or automatic? We needed an automatic so rented as we left Florence. The Hertz in Siena could not get/guarantee that for us.

Drove the Chianti road which is lovely, took our time stopping along the way. Getting into Siena was a bit dicey- we parked at our hotel and didn't use the car for next day and half. 1.5 days was enough time for us to see all we wanted to see in Siena.
I purposefully looked for and found a great hotel that had onsite parking- Palazzo Ravizza- I believe it is listed in RS guide.
https://www.palazzoravizza.com
Inside the walls with fabulous views from their terrace, easy 5 min walk to Campo, etc.

We use AutoEurope for all our European rentals. Highly recommend.
www.autoeurope.com

This may help you decide whether to pick up in Florence or Siena.

https://www.visittuscany.com/en/itineraries/a-car-tour-along-the-chiantigiana-road/

My vote is to pick up in Florence. For parking outside the walls, try Hotel Santa Caterina.
15-20 minute walk into town, or a bus stop very close to the hotel,nice staff, wonderful breakfast, good size rooms for Italy.

Edit: we also stayed in a family room, which was very nice, at Hotel Il Giardino many years ago. It too had a nice breakfast, lovely grounds and a bus very close to the hotel. Wonderful views of the city.

A car is not needed to visit Siena itself, so you could go by bus. However I would personally rent in Florence because on the way to Siena I would certainly take advantage to visit either the Chianti hills (Chianti Road SR222) or San Gimignano, or Monteriggioni (or a combination of the above). The drive to Siena is less than one hour (almost 2 hours if you take SR 222), therefore you would have time to make stops. If you go by bus, it’s just to go to Siena with no intermediate visits. Also, if you plan a day trip somewhere during your 2 days in Siena (e.g. Pienza or Montepulciano or both), a car would be nice to have.
The drive from Siena to Assisi is a quick 90+ min freeway drive. By public transportation you would spend a few hours on trains and buses.

Thank you for the information. Any hotel recommendations? I would like a hotel with outdoor seating and views. 3 or 4 star. Thank you again!!!

I posted one above Palazzo Ravizza
The terrace is gorgeous there

Hotel Santa Caterina is mentioned here often as well
Another that has views/ terrace and parking is Hotel Athena

Keep in mind the “star” ratings are not the same in Italy as they are here
They are given for things like elevator, 24 hour desk, AC, etc so a 4 star hotel in Italy might not be same standard as a 4 Star here or elsewhere

You will definitely need AC in June so prioritize that.

Booking .com is a good starting point, set your filters, read the reviews ( reviews only allowed by guests who have actually stayed, unlike TripAdvisor). You don’t have to book on booking .com, sometimes I do, sometimes I book on the hotels actual website, just depends what rate and terms I find.