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.