Great stop near Matera Italy

Are you driving? Don't know if you would consider 2h45m, but Paestum has a beach, mozzarella farms, some of the most beautiful Greek temples, and an excellent small museum. It might take a lot longer by public transportation, however.

Salerno is convenient for Pompeii as well as Paestum if you are traveling by train or bus. But again, it will take a while to get from there to Matera.

Polignano al Mare is just a bit over an hour driving from Matera, but a world apart, right on the sea. Great food! We stayed a couple of nights on a whirlwind tour of Puglia.

We also stopped in Martina Franca for lunch and decided it was a lovely possibility for a future overnight. Ostuni is lovely, too, and Lecce is just over two hours away from Matera.

Food and wine in Puglia is some of the best we've ever had at very reasonable prices.

How much time will you be spending in Sorrento, in Matera? The Paestum area is a great suggestion, and the area south of there is nice, too. So, you could go down as far as Maratea, then up to Matera--for a very Basilicata-focused trip. Alternatively, you could drive across to the northern Puglia coast.
I would look at the routes and investigate what side trips you find interesting.

There are many attractive small towns and cities in Puglia. In addition to those mentioned, you might consider Alberobello (kind of essential for the trulli), Gallipoli and Otranto. I've heard that the large city of Bari is worthwhile but haven't been there yet. I used public transportation on my two trips to Puglia and found Lecce (lovely Baroque city) to be a good base because the tourist office organizes some van trips to neighboring towns.

We had a great stop on the way to Matera. Monte Sant'Angelo occupies an incredible site on the Gargano Peninsula with views of sea and land that are staggering. Equally staggering is the history which begins (they say) with the appearance of the Arcangelo Michael in a cave about 400 AD. The cave then became a site of pilgrimage which attracted early Christians, Longobards, Normans, and whomever. Churches were built and modified in succession over the cave, and today it preserves one of the few original Lombard / Longobard constructions in Italy. Meanwhile the importance of the shrine and the location drew the need for military protection and there's a large castle with its own history. Fascinating stuff, and nearby there's an ancient hermitage with primive hermit dwellings and churches. See here and related links: https://www.wanderingitaly.com/maps/gargano-itinerary.html
Despite the fame, the town is rather uncrowded and the townspeople are still delighted by visitors.

Great suggestion, Mike. I remember driving past the castle at night, the sky was stuffed with stars and a full moon was over the castle.

I am really grateful for these fantastic suggestions. I will indeed choose from these great choices. To answer your questions, I will be in Sorrento for 7 nights and will have a car. I will now have 4 nights instead of 3 which helps a little. Again, thank you for such thoughtful replies.