The hiking season usually lasts from mid-June to late September. It depends on the snow conditions in the higher sections of the Alps, as lots of hiking routes go over high mountain passes. You should check the current situation if you are visiting in early summer, especially for Stubai High Trail, as large parts of the route run above 2000 meters of elevation.
The huts close their doors once the season ends as well, so you cannot plan any multi-day tours outside this time window.
Read more about the hiking season in Austria here.