Dugout road isn't private, there is about a section of land there maybe a little more that is public and the road runs along the southern edge. But then your off the public and into private. If you look at the BLM map or a topo map, you would have to corner jump to stay on public, may what to ask any Wyoming folks about corner jumping, as I recall its a no no and considered trespassing.
I actually hunted out in that spot last year - really ins't much for deer - not trying to keep my "secret spot" or anything but if you look at that area it just isn't much for deer. In the 2 years I've hunted 113, I've seen maybe 6 deer, 2 were small bucks and the rest were does. These were all along the south border of 113 on public. I do know some guys that hunt deer across the street in what is unit 21, along the south end. If you talk to the bio - his name is Dan he will tell you anything you need to know, he is a great guy and really knows that area well.
Good luck