They do an endurance ride there October 6-10th.  I would suspect you could follow their trail pretty easily after 100 horses trot down it.  A couple of options would be to volunteer to pre or post ride the trail. Ride management may be willing to share a map with you for the help.

Check it out at http://xprides.com/grand-canyon-xp/

I have not taken the horses down to Vegas.  But I have ridden around St George and Cedar.

The ride into Kolob arch and across Hopps valley is a good ride later when the weather cools. I used to do it for a Thanksgiving day ride when my Mother-in-law was alive and living in St George. I did it once for Labor Day and it was just too hot. So October-November for that area.

Also Red Mountain.  It's West of St George.  You ride up and look out over Snow Canyon. Both of those make great fall rides.

I would think the Grand Canyon ride you would want to do in Sept- Oct because of the elevation. The North edge of the canyon, ( which is the side you want to ride) is at 7200 foot elevation.