I assume you'll be driving, which will help you a lot since most of the places folks want to go in Puglia are fairly small. The exceptions are Bari and Lecce, to each of which I'd allocate a full day. With a car you many be able to see 4 of the small places in one day without being rushed if you don't try to cover too many miles. If you try to see most of Puglia from a single base, there may be a great deal of out-and-back driving.
I'd consider 4 full days my minimum, and I'd want to add a day to hit Matera, in Basilicata. I assume you've checked the driving time from Siena. ViaMichelin estimates 8-1/2 hours to Lecce (Bari would be a bit faster), and that's with no stops, traffic tie-ups or navigational errors.
Where will you be flying back from? Unless you can find a decent connection out of Bari, the day before your departure will see you on the road, making for the airport in either Naples or Rome, I think.