There is no great reason a hunter on public property is forced to hunt right on the property line with private property.
If you're near private property, you better be a good shot. In archery, you better be a real good shot and be over a hundred yards away from the property line.
And I bet, you weren't the first hunter looking for a deer on that private property, adjoining public lands.

I remember one state on recovering deer. If you shot a deer and it ran, the hunter who shot it last, got the deer. I had a buddy who shot a deer, it ran off, and a hunter with a 30-30 dropped it on the spot and claimed the buck. We joked about the 30-30 hunter for some years.