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If you can afford to stay in the hut, then you may well be able to afford the taxi from Täsch to Täschalp, to the summer hamlet of Ottafe at 2190m.
If not then you will have to factor in an extra 800m of ascent from the village itself, 800m below.
There is a wide dusty track you can follow from there to the hut, but more pleasant is the smaller single-track footpath that is well signposted and uncomplicated.
The hut has outstanding views and can be done as a day trip for lunch or to stay overnight. For those approaching to climb the Täschhorn you can stay there and climb the peak in a long day trip from there.
However, with its large vertical interval to the summit, it may be preferable to continue and stay in the Mittelaletsch bivouac hut far above, perhaps making a second approach day to the route itself, staying in both huts.
Equally the fittest and most acclimatised may leapfrog the Täsch hut altogether and approach in one day, from Täschalp to the Mittelaletsch bivouac hut.
If you are adopting that latter approach it may save energy to approach from the neighbouring Saas valley, using the Mittelallalin lift to 3500m and traversing the Alphubel on the way to the bivi hut.
Equally if approaching from Täschalp to the Mittelaletsch bivouac hut you can go straight up to the hut, not needing to divert via the Täsch hut. The Täsch hut itself is modern and comfortable and is a gobsmacking place to watch the sun go down.
The view on opposite side of the valley of the Weisshorn is particularly fine and is well worth the visit for this alone.