How rigorous of a hiking experience do you want? A cable car to the top of a mountain where you can walk along a ridge in good sneakers, or do you want boots and poles hiking up a mountain?
If the cable car option, I would highly recommend staying a couple of days in the Salzkammergut - St. Gilgen specifically. From there you can can take a cable car up the Zwölferhorn and hike on top. St. Gilgen is a lovely town on Lake Wolfgang. Hiking at the Gosausee is also stunning and nearby. Of course Hallstatt is on everyone's list and close by. I would recommend renting a car for the few days you spend in the Salzkammergut.