Our trip included backpacking the Centennial Trail from the French Creek Horse camp in Custer State Park to Mount Rushmore. You can follow our trail marked by the red line on the map. However my phone died after we left the Iron Creek horse camp, so I will describe it here since this doesn't allow you to edit an existing track. It will allow you to create a route.
From iron Creek horse camp we took the Centennial Trail (CT) to the Grizzly Bear Creek Trail. We continued on this trail to the Horse Thief Lake Trail, and took that to Horse Thief lake. Very nice lake and campground, along with a great camp host named Gloria. We spent Monday night there. On Tuesday we got back on the CT trail from Horse Thief lake and camped Tuesday night around the Blackberry trial that leads up to Mount Rushmore. We camped along the Grizzly Bear Creek just outside of the National Park since you cannot camp inside the park borders. We spent Wednesday at Mount Rushmore. Thursday we hiked Mount Harney, which I have saved as a separate trip.