@IScreamMonday
The first week of June makes things easier, not trickier! Most of the cable cars and other attractions will be opening for the summer season so there will be more things to do than there would have been at the beginning of May and you'll be in Austria before the main tourist season starts in July.
When people say "Tyrol" they often mean somewhere with mountains, "Sound of Music" landscapes and lakes - all of which are spread across Tyrol and Salzburgerland.
Local to where I live there are cable cars, hiking trails, zip-wires and a "Go Ape" style climbing park (one of the largest in Europe). Mountain bike trails, and cycle routes in the valley for easier cycling (bikes and e-bikes available for rent). A tree-top walk. Rafting, lake swimming, an outdoor pool complex, caves, an old silver mine. Salt mines. Paragliding. A veteran steam railway. A huge spa complex. Cake shops.
Salzburg has a castle, an Old Town, museums, chocolate shops, cafes and more cake shops and much more ...