One of my favorite routes in the park is the six mile round trip Rim Trail to the east of the visitor center. Often referred to as the Yaki Point Route, this path departs from the back of Bright Angel Bicycles, passes by the South Kaibab Trailhead, and the eastern mule barn. However it is possible to tie it in to the rest of your walk along the Rim Trail without pulling off to the visitor center or bike shop.
While it is possible to drive up to two view points along this trail ( at Pipe Creek Vista), the best way to experience it is by bicycle. The terrain is in a way where every up is rewarded with a down, with a very slight total elevation change across the 3 mile length of trail. Personal bikes are welcome in the park, and the Greenway trail system supports passage from end to end of the South Rim’s main drag (Yaki Point in the east, to Hermit’s Rest in the west).
If you don’t have your own bikes, Bright Angel Bicycles at the Grand Canyon Visitor Center will rent them to you. Reservations are recommended depending on the time of the year, you can make reservations here: bikegrandcanyon.com If you are already looking for bikes, you should consider taking a bike tour with Bright Angel Bicycles, they have a Yaki Point Tour that leaves at 10 am every morning, with a 9:30 check in. If you choose that option you will spend the next 2.5 hours accompanied by a local guide who will entreat you with a myriad of information about the Grand Canyon. From tales of adventure, plant and animal facts, history, and of course, geology. Reservations are recommended for this tour, you can make them at the link above. The tours cap out at 12 people, with 2 guides, so the group stays pretty small.