We did a rim-to-river-to-rim hike and stayed at the Phantom Ranch in January. We hiked up the Bright Angel Trail. It is almost like you are in a different canyon that the South Kaibab trail. Near Indian Gardens, we saw several deer! The creek allows for more vegetation to grown.
This trail was much more shaded than the South Kaibab. Also, there was water available at Indian Gardens. There was very packed ice near the top; I put on my yax-trax, but they didn't do much good as it was very solidly packed down. Just go slow.
For a day hike, consider either the 1.5 mile house, the 3 mile house, or Indian Gardens. We came across one hiker that was doing a day hike to Plateau Point (certainly doable in winter...it was a pleasant 40 degrees...you would want to do this with caution in the summer).
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