June will still be prime time to ski and climb on the same day on the eastside. Roads will be open to South/North Lake, Onion Valley and at least to the park entrance at tioga pass.
basecamps where you can tour from camp are a bit difficult, exp. maybe pullout spots on 120 or end of onion valley road. I'm not sure about legit car camping at the end of south lake or north lake until the officially open campgrounds.
basecamps off 395 in the forest Mammoth north are very plentiful and several good spots in the desert south of Mammoth.
you could spend many days just day touring off tioga and climbing at some of the crags on 120.
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