Long but great. Lamsjochhutte - Falkenhutte, via Hahnkampl spitze and optionally Sonnjoch. 2-3hrs to get to Reith. If you like to use your hands. Be carefull with the weather and verify the conditions of the tracks with the keepers of the cabins.
Most tours have lower alternatives.
The bus to Innsbruck you can find the first street left when you leave the Halltal.