It works. San Gimignano you can see it in the morning and have lunch there.
Monteriggioni is very small, so you can see it in one hour, including the walk from the parking lot outside the city walls. There are a couple of wine tasting rooms inside, if you like wine. I like Monteriggioni, it was even mentioned by Dante in the Inferno. Since you wil probably have luggage in the trunk, secure it with a cable lock to some hooks in the trunk, but take valuables with you.
Through the Chianti, besides the many “castelli”/“wineries”, like a castello di Brolio, or Castello di Verrazzano, the nice looking towns are Castellina, Radda, Panzano, Greve. Gaiole, just to name a few.