Traveling with two college age daughters who love quaint towns and hikes. We are driving in mid June.
This is quite difficult to answer, because there are so many quaint towns and thousands of hiking routes between the Bodensee and Salzburg. As Innsbruck is about halfway between Bodensee and Salzburg, this could be a good area. Especially the "holiday villages" in the mountains above Innsbruck could be good options. They include Patsch, Igls, Sistrans, Lans, Rinn and Tulfes, where we live. They offer beautiful views of the Nordkette mountain range and you can hike right from the doorstep of your hotel/holiday apartment. There are lots of mountain huts/alms with good traditional food scattered about these mountains. The Zirbenweg is quite a popular easy hike which offers beautiful views of the Inn Valley and mountains. Have a look here: https:/…southern-holiday-villages.html
I'll be writing about it soon for my personal travel blog, www.traveltyrol.com. You can also have a look there for some other interesting things to see and do in the area.
I hope this helps.
I don't know if you are just looking for an overnight place or whether you will be stopping a few days. The villages Linda mentioned are beautiful and unmodernized and the views stupendous-and plenty of good restaurants around. I prefer Igls or Patsch myself. If it is just an overnight you could consider Hall in Tirol, just down the A12 from Innsbruck. A very important medieval town with a beautiful old town (all up hill) and good places to eat and shop. Has some good art galleries and an old and interesting church and castle down below.
Yes, please tell us how long you would be staying in this town and confirm you will have a car?