Two one-day driving trips from Siena -- where would you recommend?

Volterra to the west and Cortona to the east. Both easy drives.

I would do Volterra and San Gimignano in one trip -- be sure to see the Etruscan Museum in Volterra. And would definitely do Montepulciano the other day. I think it is the prettiest of the hill towns in that region -- for something different perhaps work a visit to Monte Oliveto Maggiore into the Montepulciano day -- the countryside is really stunning in that area.

I did exactly what you are doing, took two day trips from Siena by car in May. (This was back in 2009.)

The first day I went south, stopping in Montalcino, the abbey of Sant'Antimo, Bagno Vignone, and Pienza, plus some other stops that were less worthwhile than those four. I regret not stopping in Montepulciano, but I lingered a little too long in other towns first. Then I drove back to Siena through the Crete Senesi.

on the second day I went north, stopping in Monteriggioni before spending most of the day in Volterra and then San Gimignano.

I recommend these two routes, even if you stop in different towns than I did. The distances were short, and it was easy to navigate from town to town. I had GPS, which I strongly recommend, mainly because returning to Siena and finding my way back to the car rental agency was confusing.

Another vote for Montalcino. Pienza, too. Monteriggione is tiny, you can walk inside the walled city in a couple of hours.