There is a dirt road that leads right to the base of Looking Glass Rock, or you can stop at the junction and hike a little bit. There is a small BLM sign pointing out the formation. Moving to the right and back along a fence, it looks like this is a double arch.
There is another dirt road that allows easy access around to the other side of Looking Glass Rock. From this opposite angle the formation is more hollowed out and I thought resembled an elephant's head.
Looking Glass Arch is one of three arches that offer short hikes along this section of Highway 191. The others are the obvious Wilson Arch and the less noticed Lopez Arch.