To split the stay between Siena surroundings and Montepulciano doesn't make much sense to me. The two are very close to each other, therefore moving base from one to the other would be a waste of time.
There would be value however to split your stay into two locations, one in Tuscany and one in Umbria, since visiting both from one location only is somewhat of a challenge, since the distances start to become too long and too much time would go into driving. For example if you chose to stay near Siena and wanted to visit Gubbio or Spoleto in Umbria, it would be a 2 to 2.5 hour drive each. I like day trips that aren't over 1 hr or max 1.5 hour each way, because driving too much 6 days in a row becomes tiring.
Which base to choose in Tuscany depends a lot on which places you want to visit. I like the area near Siena or just north of it (near Monteriggioni for example), along the freeway to Florence, because that puts you within one hour drive of everything famous in Tuscany. From that area Florence, Chianti Hills, San Gimignano, Certaldo, Volterra, Colle VE, Monteriggioni, Montalcino, San Quirico, Buonconvento, Pienza, Bagno Vignoni, Montepulciano, Cortona, etc. are all within 1 hour drive.
The area near Montepulciano is good for Montalcino, San Quirico, Buonconvento, Pienza, Bagno Vignoni, Montepulciano, Cortona. But Montepulciano is somewhat inconvenient for the Florence area as it's south in the Siena province.
If you intend to visit Umbria, the most famous places are: Perugia, Assisi, Gubbio, Spello, Spoleto, Norcia, Todi, Orvieto.
Where you want to base yourself for visiting Umbria also depends on which of the above places you want to visit. Orvieto is a bit decentralized as it is in the southern edge of the region closer to Latium, however the area around Assisi/Spello would probably provide the most central area to visit all others. Both Assisi and Spello are beautiful towns in themselves anyway. Orvieto from there is over 1.5 hours, so you might still be able to visit in a day trip from there. Or better yet, you can visit Orvieto on your way down to Rome (you can return the car there, but only with Hertz).