So... this was a great detour! The trail from Eunate to Arnotegui was very scenic and easy enough to follow. Imagine my surprise when, after our discussion and some internet research into this seemingly obscure detour, there was a giant signboard describing the route in the Eunate parking lot!
The only important intersection that wasn't marked was the second after leaving the parking lot. At the first, the sign shows Obanos and Puente la Reina to the right and Arnotegui to the left. The second fork wasn't labeled, but the sharp right takes you to the more direct and steep forest trail, and the left keeps you on the longer, gentler dirt road (zero vehicles on it when I was there).
If you follow the white and dark green posts (and sometimes posts with arrows on the top), the trail switchbacks up through the forest, follows the ridge for awhile, and then descends a bit to the now much higher road on the other side of the ridge. The
Wikiloc trail I was consulting (not the same one as above but similar) then followed the dirt road around to Arnotegui.
I, however, took the absurdly steep trail that goes straight ahead when the road veers to the left just past a large agricultural building. This route is marked with another white and dark green post and follows the ridgeline up and down several steep inclines all the way to Arnotegui. I loved being able to see the views in both directions but would NOT want to go this way with a heavy pack. It's much steeper than anything on the way up or down Alto del Perdón.
I would also NOT want to take the forest trail, and especially not the ridgeline trail in the rain, but the road looked pretty well designed and like it might be ok in wet weather.
Leaving Arnotegui via the wide driveway, you can take a right onto the trail you'll see below you and follow that down into Puente la Reina.