I've not hiked in the GC during the April/May transition. From photos that I've seen, it looks like a perfect time to be in the canyon, but my impression is that permits for that season are hard to come by.
I could easily be wrong about this, but I think that you'll need to stay flexible about your trip in order to get a permit, that you'll end up hiking the available hike, especially since your thinking of near corridor trails. I would think that the Hermit-Tonto-Bright Angel loop, as well as Indian Garden, Bright Angel, Cottonwood, and Clear Creek campgrounds are reserved.
Depending on the time of day when you leave the S. rim, the Clear Creek use area, AK9, may be accessible by spending your first night off the Kaibab, on the Tonto in BJ9, or you may be able to spend your first night a couple miles past BA campground just inside AK9. Options for exploration around Clear Creek include up or down CC, as well as East Clear Creek, although over nights are an option only up CC or up East CC. Six nights here may be a little long, depending on your taste.
You can also access AP9 from BA Campground and explore Haunted and Phantom Canyon. From this area, depending on water levels, conditions, skill, and risk comfort, I know that you can follow Phantom Canyon (wet and walled)to its confluence with BA Creek, or make your way to the Transept near Cottonwood Campground. I have followed neither of those exit routes. Again six nights may be long.
A combination of the above could give you a bit of solitude, interrupted when hiking Clear Creek Trail, passing Clear Creek camp area, and the two trips through the noise and amenities of BA Campground. A food cache would be simple and welcome, especially if you were able to immediately secure a permit and lengthen your time below the rim.
I would quickly trade a night on the S Rim for a seventh or eighth night below, and a couple of dry camp sites may allow you that option. I don't know the current water conditions.
Despite the open ended nature of your request, I'm surprised at the dearth of response to your post. There are a number of contributers on this forum, usually quick to share their considerable canyon experience and knowledge.
Wherever,including anywhere on the Tonto,your hike will be a great experience!