When the other lots are full, you might want to try the
Greens Bottom Trailhead. It's harder to find than the other parking lots (but you'll find good directions if you click the link), so fewer people go there to park.
There is a longer drive involved: if you're coming out highway 64/40, it's 18 miles from Chesterfield Mall to the trailhead.
Or, if you're taking Page/364, it's 12 miles from the Page/270 intersection.
Greens Bottom trailhead is at mile 45.7, which is about six miles from the trail's eastern end.