Hiking in Austria for 3 days

I’d rent a car and base in St. Gilgen. Take the cable car up the Zwölferhorn and hike all day. Drive easily to Hallstatt (but no hiking I can think of). Drive to Gosausee and hike around the lake.

having stayed in Hallstatt, I could think of other places to base yourself.( it is very nice but not the 'only' place) in that general area it is close to germany, so don't discount the idea of a home base in say Berchtesgaden. last year we stayed in Ramsau and hiked near the austrian border and around the lakes in the area. So much to see nearby.
The austrian 'ramsau' is also a very nice area ( just south of Hallstatt) with the dachstein mountains and nearby zip line and numerous small quaint towns.
too much to choose from.
hope this helps.

When and how are you getting to Austria?

Look on the website of the local village or region once you decide on a destination. This is a useful website, particularly for Austria and Germany: https://www.bergfex.com/sommer/st-gilgen-zwoelferhorn/touren/wandern/

For Salzkammergut: https://www.salzkammergut.at/en/hiking-trails-in-the-salzkammergut.html

In general: https://www.alltrails.com/ - need to register for a full search.
https://www.outdooractive.com/en/

Dachstein region, maybe. I did the loop thru the Schladminger Tauern in two days. You go up a laddered path thru the waterfalls to get to a lake --there's an excellent little place at the top for a snack--at the shore of a small lake. Then up to Preintallerhutte to sleep. Then up and over the top. Outstanding views across to the Dachstein. Down the other side and there is another small hutte. You could sleep there and make it three days. If you walk it in the other direction, the waterfall way is a 'one way' path, so you walk down a road and cannot go thru the canyon. Look for wildewasser weg online.

Oh and if you sleep in huts, you must have your own sleepsack. Some huts sell them, but don't count on it.

What makes most sense does depend on how you are getting in and out of Austria.

I'm at Pertisau on the Achensee again. One of my favorite places and very few US tourists. Next to the Karwendle National Park with lots to see and do. Several lifts to take you high into the mountains (Rofan, Karwendle) for hikes or hike to the Alms from below or along the lake and in the valleys. Achensee is a great lake and region at the northern end of the Zillertal.

I would suggest to go to "Seefeld in Tirol" as a base for hiking. Seefeld is situated on a high plateau about 12 miles northwest of Innsbruck. This is a mountain area, but there are no lakes nearby.

This is a list of 200+ hiking trails in the Seefeld area:
https://www.seefeld.com/en/walking/hikes

You can choose by lenght, difficulty, and altitude difference.