I'm not sure about that particular trailhead, but in general, you'll get a ride sooner if:
You're one person.
Female.
Smiling.
No sunglasses.
With a sign clearly stating where you're going.
And it's someplace close.
It's a time other people are finishing their hike (i.e. Driving away)
And you chat up people at the trailhead before they depart.
Multipurpose your tyvek ground cloth and pre-write "Park HQ" on one part and "Lipan Point" on another section.
Could you bring a beater bicycle along? If starting in the am, I doubt many other people would be finishing their hikes then. Unless maybe it's a spot photographers go for the sunrise? Peak season, during daylight hours, there's a shuttle bus to and from the South Kaibab trailhead. I don't know if there is shuttle service to other trailheads.