Well, how far from Basque Country are you willing to go? I’ve spent time in San Sebastián, but don’t know the region too well. You mentioned Bordeaux…you can rent a car, and drive up the southwestern Atlantic coast, if that interests you, or go in the other direction, go westward along Spain’s Atlantic Coast. Further east, in south-central France, there’s Carcassonne, a wonderfully-preserved (if slightly altered in modern times) medieval castle/walled city. La Rioja is nearby, you can definitely explore that region.

I agree that not much is said about northern Spain. It is not as touristy. For scenic beauty, this part of Spain is hard to beat, and we ate very well, too. What we did after we left SS was follow the coastline west, eventually going all the way to A Coruña (Galicia).

We wanted to visit Bayonne but decided not to.
One thing you might consider is going down to Pamplona. Rioja is a good idea.

We were in San Sebastian at Christmastime. Loved it.
It was such a beautiful city, in a gorgeous region. The St Thomas fair took place while we were there, and we saw many men, women and children dressed in traditional Basque costumes. You’ll see the ETA signs (Euskadi Ta Askatasuna). And signage will include both Spanish as well as the Basque language, Euskara.

What we did was fly into Toulouse (so sad what’s happened there, it is another lovely city).
Carcassonne is only an hour away from Toulouse.
We did Toulouse>Pau>Hendaye>San Sebastian. Pau is a nice, walkable town with Henri IV’s chateau.
One place we wanted to visit but couldn’t was Oloron St Marie. With a car, you could do this.

In San Sebastian, we stayed at the Pension Bellas Artes and I highly recommend it. (But maybe you already have lodging figured out?) Leire and Carmen took very good care of us, giving us a map of old town (with all the best pinxtos) and telling us where to go after we left SS. If you stay with Leire, she will tell you all you need to know about day trips. You will definitely want to drink cider and eat pinxtos. There is a wonderful Fellini exhibit at the San Telmo museum.

One nice day trip you might take is to the seaside town of Getaria (half an hour west), we had a wonderful meal at the restaurant right on the harbor. Further on (and inland) is Guernica, with its sad story, then Bilbao, you can’t miss Jeff Koons’ guardian dog in front of the Guggenheim, and still further, about two hours, is Santander (Cantabria), another larger seaside town. A half an hour west of Santander is Santillana del Mar, a preserved medieval village (a UNESCO site) which still has working farms around it. It’s kind of touristy but not any more so than Carcassonne, we liked walking around that picturesque spot.
A bit further west, still going along the coast, is yet another seaside town that we really enjoyed visiting: Comillas. This town has one of the few Gaudi buildings outside of Catalonia: El Capricho. We happened to visit this amazing home when nobody else was there. After that, we stopped at the cemetery on the outskirts of town—very atmospheric. Comillas is about a 2 and a half hour drive from San Sebastian.
A bit further is Llanes, another lovely place on the water, interesting buildings and beautiful harbor, and further still (4 hours from San Sebastian) is Oviedo (Asturias), where we spent Christmas. We had a great time in Oviedo, a city that is very proud of its Woody Allen connection.
The Picos de Europa is a mountainous national park well worth visiting, and there are caves very close to Oviedo. But as I said before, this is more like 4 hours away from San Sebastian.

Hope this helps. I might try to post a few pictures.

Thank you! These are great suggestions! I have some new leads for my research now! Thanks again!

I am not familiar with this area but ran into some information about this scenic train route…
Look into the train from Bayonne to St. Jean Pied de Port. It’s supposed to be a very scenic route — think there are some references to it on trip advisor. St. Jean looks to be historically interesting. It is on the pilgrim’s route (not sure how you follow that, definitely NOT by train) to Santiago de Compostela — one of the “cultural capitals” of Europe, although perhaps too far from your base to explore on this trip.

I did not mention the pilgrim’s route to Santiago de Compestela because, as Oldlady says, it is probably too far away, but it is an interesting tradition, you will definitely see references to it on the northern Spanish coast (and in France), and Santiago is well worth visiting.