Planning to hike in the Innsbruck area in the last couple of weeks of May. Looking to take a cable car (i.e. Nordkettebahn/Patscherkofelbahn, etc) up and hike something like the Goetheweg, Zirbenweg, etc. Which of these will be accessible? (Or any recommendations for hiking around this time?) Where is the snow line typically at around this time?
Hiker's paradise is the town Seefeld in Tirol high above Innsbruck, i.e. in a plain having an elevation of 4000 ft, providing 100+ hiking routes of different length and severity.
https://www.seefeld.com/en/hiking-trails.html
Seefeld can easily be reached by a scenic train ride of 36 minutes from Innsbruck in case you prefer to stay in the city.
I would second the idea of Seefeld. I was there in October. It is small very nice town with great hiking. You may still have quite a bit of snow in May, so keep that in mind.
Well, there's the Stubaital. You could even ski on the glacier if you wanted to. Bus and train (Stubaitalbahn). You need to check the lifts that are running and conditions.