Renting a villa in Tuscany

Never rented a villa. You don't say how many people are in your party. VRBO shows many rentals, with or without private or shared pools. A smaller house with 2 or 3 bedrooms may be identified as a villa.

I like this site, used it for Germany a couple of times:

https://www.traum-ferienwohnungen.de Type in 'Toskana'. Filter out bedroom numbers, amenities etc.

Some agriturismos have small cottages on their properties, some have a few individual apartments. Don't know if you are travelling with children or not, If not, consider separate apartments on the same property. We stayed here, a great little place with a lovely host. She's added a pool since we were there: https://www.fontecastello.it/

Also, https://www.agriturismo.it/en/

Take a look at www.montorio.com just down the hill from Montepulciano. There are 5 very elegant apartments in the villa that might answer your needs. We enjoyed the place very much on our own trip a few years ago.
An added benefit (for us anyway) was the free, secure parking on site and a handy location that made it a snap to get out on the roads in the morning to explore the countryside.
It's a short stroll away from the very pretty little San Biaggio church, and one of the best restaurants we found was located just across the street from there.
Might add that there's a handy local market not far from the villa if you want to do a bit of self-catering.

I’ve rented villas twice (Umbria and Tuscany) from Tuscany Now and More. They have a clunky name but good and reliable service. It’s a smaller outfit and they have exclusives with the properties they represent, so you won’t see the same villas elsewhere. They were very helpful with rebooking a covid-time booking several years later and honoring the original price.

Worth a look anyway. Not the cheapest but they have varying sizes and a la carte services (such as cooking). We used cooking services at both locations (not every night) and were very pleased.

Here is a smaller villa that is attached to the bigger one we rented in Umbria. I can vouch for it as we had a tour and really enjoyed the owners. Casa del Poggio