It’s a bit west of Palermo, and I don’t know if there are organized day tours available, or if you’d be renting a car at all, but a trip to Sicily should include Segesta.
Sicily, as you may know, was called “Greater Greece” by ancient Greeks. First-born sons inherited the father’s land, so younger brothers had to look elsewhere for land. Sailing to what is now Sicily provided that, an extension of the Greece homeland. At Segesta there’s an ancient Greek temple, frankly in better shape than what’s at the Acropolis in Athens, plus the remains of an ancient theater, uphill from the temple. Short of making an extra trip over to Greece on your Italy trip, it’s your best chance to see such treasures. Actually, Agrigento, which is at the southern coast of Sicily, has even more Greek temples, but then that’s farther away, and would call for even more time. If you can get a little bit away from Palermo on your visit, it will be memorable!