My curiosity getting the best of me, I decided to head that direction. Then, I found another sign
I'm guessing "Old Jefferson Road" is the old Jefferson Highway, US 71's predecessor? I've just recently taken an interest in the Jefferson Highway, so I'm still piecing things together.
It appears to closely parallel US 71 though it shifts and runs through Kansas, Oklahoma, and Texas eventually ending up in New Orleans. It's possible there was an auxiliary route that passed through Arkansas, but I'm still looking for information on that.
I hear the Powers Museum in Carthage, MO is involved in doing some research, so I may have take a couple days and go up to see what they can tell me.