If you were on a tour, you have probably already been to Florence. Although Lucca is "on Rick's list", it's a bit west of the towns most often discussed on this board. Google Maps suggests that Greve in Chianti (picked randomly) is two hours away by car.
We visited Lucca as a train daytrip, I think from Florence. It is certainly worth sleeping there, but it's a different trip than (a later trip we made, for hill towns ... ) sleeping outside Sinalunga and driving an hour or less to a dozen popular towns. You have to hope for a reply from a specialist in western towns.
Other posters are absolutely right that you need a car to make the best of this visit. However, I am sure that driving in Pisa is no pleasure. I found parking for Siena to be no pleasure! You need to use the search box top center to learn about IDL, ZTL, and insurance in Italy. I am not a lover of "having a car" (neither of my parents ever owned a car in their entire lives), but I would not visit Tuscany without one. Do not treat the car posts as a distraction from your planning; they are a crucial part of it.
Remember that the better wineries require reservations. We chose to go to ones that had high-end restaurants associated with them for (prepaid) lunch, and had only bar-snacks at our hotel for dinner, after a dip in the pool to cool off. (Check October climate online before taking a home without air conditioning.)