Only one we've stayed at that might fit, but I'll lay the logistics out as it may be stretching your walking limits
Il Residence Le Meridiane è composto da 20 appartamenti e domina un uliveto situato nella città di Siena sulla collina dell’antico convento dei Cappuccini. Vant
www.lemeridianesiena.com
Le Meridiene is a moderately sized gathering of purpose built apartments, with some facilities (giant chess, swimming pool, laundry, bbq area and pleasant grounds with views etc.). Apartments modern, if (for us) a little soulless.
It's a 35-40m walk from the train station, or a modest taxi ride.
It's set near a small hamlet, on a small hill looking towards Siena, and is about a 50m walk to the campo, but there are buses about 3-4 min walk from the apartments. A taxi home from the city in the evening was very modestly priced.
Good if wanting to feel relaxed and away from the crowds, with good facilities, but less so if looking for proximity to the city, wanting to be very active, or wanting to feel in the Italian vibe. It was too soulless for us - out of choice we were renting apartments long before airbnb came along. So a poor (but not bad) choice for us, but might be much more to your needs.
Another possibility might be Chianciano Terme. It's on the mainline route, so well served by trains, and being a spa town, might be blessed with pools etc. It's a shortish bus ride from Montepulciano, which drops you off at the shallower end of the city to make walking in easier (you can get dropped off earlier, for a steep but faster climb).
There are a couple of very nice places we've stayed at in Piemonte and outside Verona, with lovely pools, and a more humble place with pool in Ravello (but with a lovely garden setting with stunning views and the waft of wisteria) but I've stuck with tuscany as mentioned.
