Ten day stay in Tuscany in October - Stay in one town or two?

Yes to both Siena and renting a car! A car is really necessary to explore Tuscany and its small villages. Siena is well positioned for exploring Tuscany. The Chianti villages are just north of Siena(Radda, Castellina, Panzano, Greve etc)
Or for another location, go to Lucca (west) or Cortona( east) and use them as bases to explore another part of Tuscany.
Siena, Panzano, Castellina- we rented houses or apartments. Radda, we stayed at Il Borgo di Vecine, overlooking vineyards, charming town. Look at the dates for the Palio in Siena( June/ August), a fabulous experience.
Just below Cortona, we stayed at Il Falconiere, fabulous dinners at this lovely hotel set in the vineyards.
Lucca- stay inside the walls at Hotel Palazzo Alexander. They parked our car for us in their lot where we left it while in Lucca.

I suggest 2 bases, one more northerly, the other to the south. And yes, rent a car. Tuscany distances are deceptive. What looks like a short jaunt may take a long time due to the narrow, hilly, winding nature of the roads.

A car is desirable for any visit to Tuscany. You should choose a base near some of the places you want to visit. We chose a rural luxury hotel, because we didn't want to deal with city traffic and parking after a day out and about. Note that some of the popular hill towns have long walks from the parking lots, or long waits for a shuttle bus, if you're lucky. You need to do some research, while thinking realistically about how many miles of walking he will have left, after (for example) a steep hill up from the lots in Sam Gimigniano.

Check the climate reports for October. We aren't even "pool people", but we relished a dip in the hotel pool after a sweaty day of touring in July. Usual warnings about IDP, ZTL, Italy insurances. (Search box top center of this page.)

I agree that Siena matches your mention of a "center city", but this is not typical of the hill towns of Tuscany. There's a big spectrum between Siena and Monteriggione. (The latter has limited parking, but October might be OK.) Orvieto is not as big as Siena, but has a lot to do in one place. We tried to do three hill towns a day with a car. We prioritized some luxury winery restaurants for lunch, and had a tiny dinner at the hotel bar, because of gastric reflux at bedtime diagnosis. Had to pay in advance and reserve every winery tour.

I also suggest 2 bases, one north and one south. Definitely rent a car. If you want a suggestion for a Siena hotel with parking just let me know.

Yes to 2 towns, yes to renting a car
You really need that car to get to most towns to explore.

I’d pick one base North-Siena or maybe Lucca- Siena is larger and busier and steep
Lucca is flat but a bit further from most well known hilltop villages

Then a southern location in Val D’Orcia region- we like both Pienza and Montepulciano
Opted to stay in Pienza for our second visit because it’s flat, very easy to walk around. Easy to come and go with a car and parking not much of a problem.
Montepulciano is very steep and a lot busier with tourists

You first need to decide what you want to visit before you can decide on the one or two locations.

A car is almost a necessity, or you will be severely limited on the towns you can visit. Siena is a great central location, but it's a city (not a small village) with a large ZTL (Restricted Traffic Zone) therefore you need to be comfortable driving in a city and you must choose a hotel that is well positioned for car use (and preferably with parking). If you decide not to rent a car, then Siena is best because it's the transportation hub for its province.

I am not an academic authority on this topic, but note that the Renaissance came late to Siena. That makes the (excellent) art there tend to be an older style than it is, say, in Florence.