Turin I have personally pushed as short break spot several times in this forum. I really like the cinema museum, which is an outrageous building that was intended to be a synagogue.

Bologna is another one I've pushed, Genius1 has a photo rich thread in the Trip Report forum. It also has fabulous train connections to dozens of smaller Italian cities which makes it an efficient way to get to all sorts of amazing towns and cities.

So some other less obvious ones on BA's network:

Luxemburg for the bus or train to Trier.
Bastia
Catania
Cagliari
Quimper
Zurich - particularly in summer
Ljubljana for Lake Bled (direct bus service from the airport, 4€)
Zagreb
Vienna for Bratislava

Then there is Iberia domestic hub out of MAD and to some extent BCN. There are dozens of great spots which are easy to get to, and efficient with Avios too. So Asturias, Santiago de Compostela (St. James of Compostela), León, Pamplona all spring to mind. I also like San Sebastián, also called Donostia. Sea front below.