So many beautiful towns to choose from it's difficult to decide. Here is my attempt. All opinions from fellow travelers and/or locals are welcomed and thank you in advance for taking the time to respond
Day 1. Arrive in Bari. Stay one night and hopefully recover from jet lag
Day 2.,3, 4 Take train to Brindisi and pick up car rental
Drive to Lecce and stay 3 nights with day trips to?
Day 5, 6 7 Drive to Martina Franca. Stay 3 nights
with day trips to Alberobello, locorotondo,Cisternono, Ostuni, Gallipoli
Day 8, 9, 10 Drive to Matera. Stay 3 nights
Day 11: Drive to Bari, return car rental, stay 1 night
Day 12: Depart 6 a.m. flight to the States

Note: picking up car rental in Brindisi rather than Bari to give more time to adjust to jet lag
Would love suggestions on day trips from Lecce
Any hotel suggestions for last night given that very early flight out of Bari?

Are you staying in Brindisi days 2 through 4? Or is it Lecce? I'd actually consider picking up a car in/around Lecce instead, assuming you keep the same order.

On your plans for Lecce itself, you might not have enough time for day trips. I had 4 nights with day trips, I could have easily done 5 to spend more time in Lecce itself. It's small, but has a lot to offer. Edit: I did Ostuni and Otranto from Lecce.

Personally feel you have too much going on out of Martina Franca. I felt 2 nights was sufficient for Matera for my tastes, ymmv. One suggestion might be to actually start in Matera upon arrival in Bari - you do not need a car for Matera (and really cannot drive in the old city anyway).

Maybe check out this recent thread for other ideas/suggestions: Puglia solo trip

i second Travel-Nerd's suggestions about starting with Matera and giving it 2 nights. That said, if you want to hike in the area or are using it to recover from jet lag, a 3rd night might serve well.

Your proposed daytrips to Gallipoli and Otranto are best done from Lecce.

I agree to take the bus from Bari to Matera on arrival. Then bus back to Bari and rent car at airport, or stay night in Bari and rent car the following day...

Some of those towns can be visited on the drive from Lecce to the re-location spot in Martina....

Certainly visit Gallipoli from Lecce, not from Martina.

Could have lunch in Ceglie on the transfer day from Lecce to Martina....far less touristed than your other proposed towns...and very good for restaurants..

We've stayed before, in Bari, at BRA HOTEL. I debated about staying again next month because the price has increased a lot, but it's a very good, very small hotel and has a very distinctive style....I don't know what to call the style, but think lots of black marble with gold accents.....the polar opposite of what you'd think of when you think of Puglia but because it's such an outlier, we found it compelling,, even though you'd not think so from my description. For no reason I can think of, they gave us a free dinner in their restaurant.. Breakfasts are lavish and outstanding.

We decided to stay in Murat, rather than Bari Vecchia; those would be the two main areas for hotels so maybe consider which one appeals more to you and only then,, look for hotels in your price range....

I agree with going to Matera directly. I don't think three nights is too many if you enjoy exploring. There are quite a few excellent restaurants in Matera as well. You could then go from Matera to Martina Franca. Then to Lecce.

I picked up my rental car in Matera from Avis and ultimately dropped it off in Lecce. I kept the car while I was in Lecce so I could easily day trip. Lots to see in that part of the Salento and the driving is easy. Took the train from Lecce to Bari, stayed in Bari and then flew out from there.

So that is an option too.

ETA: In Bari I stayed in Murat at the JR Hotels Oriente Bari