Our 2 separate trips to Basque Country were one just in Spain (with rental car), and then in France this past July (arriving in Bayonne by train, then renting a car in Biarritz when leaving a few days later). Bayonne is scenic, has an interesting cathedral/cloister, the vibe by the Nive River is festive, and the Basque Museum (Le musée basque et de l'histoire de Bayonne) was worth a visit. Excellent chocolate in town. Ice cream, too, in unique flavors. Not sure how fast or easy it would be from St. Sebastián to Bayonne by public transport.
Maybe virtually impossible to do as a public-transport daytrip from St. Sebastián, but Espellete, where AOP peppers are grown (and the red chile is used in things from curing Bayonne ham, to chocolate, to cheese, etc.) was a very fun stop in July, the day we checked out of Bayonne and got our car. In October, there may or may not still be chiles hanging outside every building in town, covering the white, half-timbered places with deep red.