Start from the South Rim of the Grand Canyon and traverse the Tonto Plateau in the Grand Canyon? If it’s not Christmas-New Years, my understanding from the backcountry office is that in normal years, while the main trail corridor (South Kaibab, North Kaibab and Bright Angel) can be be impacted; other trails have lots of availability in the winter (and, seeing I was Alaskan, they’ve commented that they get a lot of Canadians and Alaskans in the winter).
Generally, you’d be below the snow, and have lows in the 30s, but you’d want to be prepared to be dusted and for nights in the 20s.
Grand Canyon has been my most common January trip for the last decade.
Last I saw, due to covid, you needed to get your permit online, but if you had your printed out permit, that was all you needed. (And, yes, backcountry rangers are still checking for permits).