If you're in good hiking shape, a solo hike to Bright Angel Campground and back to the rim on the following day would be both safe and memorable.
As you probably know, South Kaibab is steeper, shorter, and more visually panoramic compared to the greener, wetter inside passage of Bright Angel Trail.I would decend Kaibab to BA campground, spend any extra time and energy walking N. Kaibab through the narrows and beyond.
Bright Angel Trail between BA campground and Indian Gardens is short, relatively flat and fast. Indian Gardens is a great oasis in the heat,and intersects the Tonto Trail and a short trail out to a Colorado River overlook. I don't recall the name. The overlook is nice, maybe anticlimatic after the Kaibab, but a distance on the Tonto will provide another impression of the canyon. I would save about three and a half hours for the steeper hike from IG to the rim.
You could also connect S. Kaibab and BA via the Tonto on the way down or up.
I would take sticks for the Kaibab, prepare for heat, and plan on wishing for a longer stay.\