Things operate differently during "peak (reservation) season" than during the "walk-in season."
There is no self-service option during peak season. Late arrivals are provided with a list of available sites (posted on a bulletin board) and instructed to report to the campground office when it opens the following morning.
See when you reply with more than a one or two word answer we know what you are talking about. Yes you are correct that at peak times you check the list, setup on an empty spots, and then pay at the station in the morning. I believe the point that was trying to expressed to the OP was that you still have to pay even if you arrive after dark, but at least it is possible to get a front country site without reservations at the last minute assuming spots are available. Also, I don't think you would be out of line to check the list, pay with an envelope in the box (which is at every campsite), and setup camp. I doubt the rangers would be too upset you didn't pay them in the morning and instead put your fee in the box as long as you setup on an empty site. Especially if you plan to leave camp early.
Pretty much your only option if you both arrive late AND leave early.
Otherwise you are in violation of the regulations that require you to register and pay for your camp site.
But it makes sense that they don't include instructions for a situation that occurs very infrequently on signs designed to help move people through the check-in process. They obviously don't have room to print ALL the rules and regulations.
Great theory. Except that there is no "box" at Elkmont during peak season. There is no mechanism for paying the camping fee except in person at the campground office, during normal working hours.
The "box at every campsite" is completely a figment of your imagination. The only thing at every campsite is a post on which to clip one's registration form, which can only be obtained as noted above.