The Rüsselsheimer Hütte is the perfect refuge for this trip. It offers space for about 30 people and is a very charming refuge. It's lying at 2323m, it takes something between 1h and 2h to get up to it from the parking spot.
Here the way branches. Follow the north way to the western ridge, the marker shows "Hohe Geige - Westgrat". The west ridge is the more beautiful path. It difficulties are mostly UIAA I, a few spots II. To the east you can follow the normal path, but better descent using it.
Enjoy the wonderful view. Ortler, Bernina, Weildspitze, Weißkugel, Hintere Schwärze, Similaun, just to name a view summits you can see. There is nothing blocking your view here, if you have luck with the weather, it's just amazing.