The day after the Arlberg Giro we went running in the hills above Sankt Anton. We jogged up to the Leutkircher Hut, grabbed soup and an apple strüdel and then ran along a bit and dropped back down into the valley. It was 17 km with approximately 1200 m of climbing. Not bad.
It looks like it’d be possible to sleep over at the hut if you felt the need to.