In reply to VictorM:
> taking public transport from Innsbruck to Hammersbach currently takes more than three hours so that pretty much rules out doing it in a day.
Indeed. I've never done the Höllental route myself, but whenever I've been in the area in the wee small hours the Hammersbach carpark has been pretty busy around 4 or 5 in the morning. And they can't all have been greyhounds like my mate who did Höllental and the Jubiläumsgrat in a day.
The options from Innsbruck are: train the previous afternoon & hut, train the previous evening and b&b in Grainau or Hammersbach, or drive early in the morning.
The railway might be open again by September, but even then it's not going to be that much faster.