I don’t know what route Rick Steves suggests.
There are basically 3 ways to drive to Montepulciano from Siena, all 3 are anout the same distance:
1. the shortest in time is through the Siena-Bettolle-Perugia freeway to Sinalunga, then up to Montepulciano. This one can get you there in about one hour.
2. The other is through the via Cassia (SS2) to Buonconvento-San Quirico, then up to Montepulciano via Pienza.
3. The other one is through Asciano.
No. 2 or 3 are only about 20 min or so longer than no. 1.. I like no. 2 best, but if Rick Steves talks about the Crete Senesi, maybe he’s talking about Asciano, or also 3 via Buonconvento.
Neither San Gimignano nor Lucca are on those routes, as they are both in the opposite direction, so they would be a major detour. In the case of Lucca that would be too crazy of a detour to even consider, as Lucca not only is in the opposite direction but also very very far in the opposite direction (2 hours from Siena). San Gimignano is in the opposite direction also, but it is close enough to Siena (certainly way less than one hour from Siena) to make it doable on the same day.. you’d just need to go to San Gimignano (less than one hour away to the north of Siena, then back track back south to Siena and proceed to the original route to Montepulciano. You should be able to drive from San Gimignano to Montepulciano in less than a couple of hours, assuming no stops (which you will certainly make for some photo opportunity).