Anyone know of a good agriturismo or country house in eastern Basilicata or western Puglia?

Check on Booking.com by the specific nearby city's name. Virtually every farm in the area will have rooms or apartments for a secondary source of income. TripAdvisor.com is another place to locate agriturismos.

Another way to find rental properties is to go on Googlemaps and put in a specific city's name. Then hit the "+" key two or three times to magnify the map. Up will pop icons for various businesses--including hotels. Often they're going to have their direct hotel/agriturismo websites to click on..

Charlene from Denver, who often posts here, recommends a place. I'm taking the liberty of copying one of her replies to someone else and posting it for you, since she hasn't! (Charlene, hope you're well.) "Rent a car when you leave Matera, and drive to a masseria in the Val d'Itria for 5-6 days. We spent 6 days at the beautiful Masseria Montenapoleone near Pezze di Greco. It had a pool, and the weather was still warm enough Oct. 17-23 for us to use it. They also have a wonderful, free communal kitchen for guests to use, stocked with olive oil and other necessities, plus fresh fruit from their orchard. Wonderful breakfasts. We absolutely loved staying here too! From here we drove to Alberobello, Ostuni, Locorotundo and also enjoyed visiting Torre Canne, a small coastal town. There are many masserias in the Val d'Itria, and I highly recommend you stay in one."

Thank you both for your suggestions!