Badami is the natural base for the Chalukyan temple circuit of northern Karnataka. Pattadakal and Aihole are the obvious day trips, though they do not yet have full World66 pages; pair them with Mahakuta and Banashankari if you have a car.
Bijapur is farther north and belongs to a different historical world, but the contrast between Adil Shahi domes and Chalukya sandstone is excellent. Hampi is too far for a comfortable day trip and deserves its own stop.