Jaén is well-placed for the Renaissance towns of the province. Úbeda and Baeza, 60km northeast, are a paired UNESCO World Heritage Site — small towns full of Vandelvira's churches and palaces, with serious tapas culture. Go for the day or stay a night.
Granada is just over an hour south by bus — easy enough to do as a long day trip, though the Alhambra deserves more time.
The Sierras de Cazorla, Segura y Las Villas is the largest protected area in Spain, two hours east. Hire a car and aim for one of the small villages within the park (Cazorla itself or Hornos) — proper Spanish mountain landscape, with vultures, wild boar, and Spanish ibex.