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I'm heading to Wyoming next week (6/13) and wanted to see that view of the Tetons from Table Mountain.

What are the conditions (namely on the switchbacks), and are crampons/snowshoes necessary?

I haven't been on that side, but things are melting out very fast on the other side, and it hasn't been cold overnight. The standard route on Teewinot was snow-free to almost 9500' last Sunday, and some hardy souls hiked to Amphitheater Lake in flip-flops. It's dry this year, especially lower down.