This gentleman, Pika, provides sound advice to carry lots of water for your June trip.
Above Cottonwood Camp on the North Kaibab Trail in June you can get water at the pumping station, Roaring Springs, Supai Tunnel (and from the river to Roaring Springs you can filter from Bright Angel Creek).
Throngs do the corridor and to a lesser density other GCNP trails in late May through early September (but I don't anymore).
I bet you will want to try the rim to rim again during a more seasonly appropriate, less crowded time of year after this trip. On that next visit you can say you did it when it was brutal - which in some circles is priceless.
May through September's is a fantastic time of year to really enjoy floating the river.
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