"The Arizona Trail north of the canyon in the Kaibab Forest is excellent."
On the North Kaibab National Forest map, the AzT is on their #101 trail. That is a popular single track mtn bike trail since it is in the national forest outside of the national park. Early in the season, there will be deadfall trees on the trail.
There isn't much there, just extensive solitude (antonym of the South Rim Village), big meadows lined with aspens, some Douglas fir and Ponderosa pines, spruce clusters that are excellent windbreaks for campsites, cool summer weather typical of 7000-8500' altitude, noisy turkeys and big mule deer. Brachiopods (fossil sea shells) are seen in the exposed Kaibab limestone. Those shells are now 8000 feet higher than current sea level. See what happens if you stay there too long.