Thank you for the correction; I was under the impression that even day hikers needed permits.
Yet Grand Canyon is a very big place and there are lots of opportunities for solitude, away from crowds - like caves. You rarely hear of them, but the Redwall Limestone is full of them, the vast majority undocumeented and unexplored. One night in camp, while exploring caves at the behest of the NPS, we calculated that, given the length of Redwall cliffs in the Canyon, and estimating about 1/2 mile of cave passage per cave, there was perhaps more total cave passage in Grand Canyon than in Mammoth Cave. Speculation, true, but quite reasonable....
There are plenty of "unmaintained" trails in Grand Canyon - basically they are the equivalent of normal trails elsewhere, and I have never had to fight for right of way on any of them - Grandview, New Hance, Old Hance, etc.
And there are less visited NPs - like Gates of the Arctic. Last I looked, it got 28,000 annual visits. I suspect it has no entrance fees. There are lots of hidden gems tucked away in the NP system that are off the beaten path, tricky to access, without a local Chamber of Commerce beating the drum.
one or more gifts or other benefits from the co-op.