While trawling around the internet looking for ideas I came across Hotel Prategiano in Montieri, Tuscany. This looks very promising as a centre from which to do day trips by bicycle, or by car if we want to go further afield. Their rates seem quite reasonable and they are bicycle friendly. There is some information about the village of Montieri itself here.
There are other possibilities for Tuscany too, such as self-guided cycling tours:
and I am sure there are probably many more. The problem I see with these sort of tours though is that they are 'fixed itinerary' so if you wanted to visit one place more than once it makes it difficult to do so. Given the nature of the Tuscan region, then, I am more in favour of using just a single base - or perhaps a couple of bases - in order to be as flexible as possible about daily destinations.