Unless you're committing to having a car for at least some of your stay that will greatly limit your range of choices. Picturesque hilltowns are not generally well served by mass transit and make day tripping difficult.
This is a pretty good guide to towns accessible by rail or rail and bus.
https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQnWsr0iSSHPVvb-rQFbVCE3S5KdYhlWY2QDBzls7PBzA&s=10
If you want to go by rail I would suggest Arezzo as only moderately touristy and connected by rail to Florence - the local transit hub - that puts eastern places like Cortona (train and bus) and Lake Trasimeno within reach. Or Siena - very touristy - but well connected by tours, bus and train to the areas around it and Florence as a hub. Both are worth exploring for a couple days on their own, have many good restaurant options and car rental agencies etc.
People choose Lucca because it is on the flat and is connected by rail to towns around it because trains work well on flat land. And while it has its quiet charms it is not a Tuscan hilltown of photo and film like Volterra, Montalcino, Pienza etc.
If you really want the hilltown experience I think Montepulciano is the largest of the famous southern towns because at three weeks I guarantee you will have eaten at every place in a small town at least 3 times by the time you're done. I would choose the largest charming town with the most options and go from there. There is no town perfect situated to allow easy access to northern towns - Volterra - and the southern towns. In the northern area you might look somewhere like Castellina in Chianti which was small but trying hard to get on tourist map when I stayed there many years ago. It is well positioned between Siena and Florence to visit the northern areas.
Buonconvento is really well positioned with bus and train access to Siena and easy car access to the southern towns and a bit far for Greve and the other northern towns but doable. But the town itself is really small so an agriturismo outside could be a good option but is a different trip than you are describing.
Good luck,
=Tod