From what I read, when they started seeing signs to that effect, they decided to turn around and go back to the interstate. The road was too narrow at that point to turn around so Kim backed up for a while until he saw a road turnoff (this was the spur road that is usually locked by a gate in the winter) and took that, probably thinking it would loop him back to the road or that he'd find a place to turn around. They got stuck in the snow on that road.
What you read is wrong. The first two signs are on the interstate.