In reply to doniceti:
I've used the Karwendeltal in late June before now (just N of Innsbruck). A number of the key passes are a bit lower than elsewhere in Tirol so more likely to be free of snow.
There is also a lower level walk, "via della pace", I think, which runs along the old front line in Trento that again runs at a lower level.
On a smilar basis, Arco should give excellent climbing and some via ferrata, and plenty of camping. Early season too so should be pleasantly warm but not too busy.