From Innsbruck, you can take the J bus to Igls (Patscherkofelbahn). To save 1000 m of elevation gain and 7 km of walking, we took the cable car to the Patscherkofel mountain station. The route begins from the Patscherkofel on the Stone Pine Path (Zirbenweg ) to the Glungezer Hütte alpine hut (2,610 m), and along the “Glungezer & Geier” ridge route across the seven Tuxer summits to Lizumer Hütte alpine hut (2,019 m). The section between the Glungezer Hütte and Lizumer Hütte is marked to take 7.5 hours (and it actually does take this long). The ridge consists of large boulders rather than an actual trail for the most part until Naviser Jöchl.
From Lizumer Hütte, the designated trail follows the central alpine route 02A over the Grafennsjoch to Weidener Hütte, from there over the Nurpensjoch to Rastkogel Hütte alpine hut (2,117 m), to Hochfügen and over the Loassattel to Kellerjoch Hütte alpine hut, before descending to Schwaz.
We decided to do an alternate route from the Weidener Hütte, taking us around to Untere Nurpensalm and over Kleiner Gilfert (part of this trail is relativley new and is not visible on the Outdooractive map) to Rastkogel Hütte.
Since I had catch a bus from Innsruck, I knew that I would not be able to hike the 30 remaining Kilometers down to Schwaz. Instead, I went back via Kleiner Gilfert and down towards Weerberg. Once I hit the road with traffic (after about 4 hours of walking), I stuck my thumb out and a kind woman picked me up and gave me a ride to Weer. From there I took a bus back to Innsbruck.
Note: The first 2 days of this route are normally done in 4 days. We skipped every other hut making it a pretty tough hike. Good fitness for hiking is necessary, otherwise I would advise stopping overnight at each hut. You can see the recommended Inntaler Höhenweg here: https://www.tirol.at/reisefuehrer/sport/wandern/wandertouren/a-inntaler-hoehenweg