I have a Goodtime Classic II, and I think you're on to something regarding the coordinator rod being too tight.
It's an easy fix:
First, loosen the strings.
Then loosen the outside coordinator rod nut until it doesn't touch the rim, Do the same with the inside nut. This should allow the rim to move into a neutral non-compressed condition.
The rim may be slightly warped from being compressed for a while. If so, tighten the inside nut to bring the rim back into round. (You can use a ruler to check roundness. Measure top to bottom and side to side. Both measurements should be the same.)
Now snug the outside coordinator rod nut to the rim.
Do you mean the tension hoop appears to be out of round (not concentric with the rim) or it's pulled down more near the neck and tailpiece? If it's the latter, I'd loosen the head all the way around and start pulling it down evenly. Hopefully the tension hoop is not bent, but even that is easily remedied, especially on a Goodtime. Using something like a file card on edge, make a pencil mark to indicate the height of the hoop above the head. Work to keep that constant as you tighten the nuts back down. If you're unsure about how tight to go, look up Steve Davis method. If you place a six inch ruler at the end of the bridge, a dime should just pass under it. With a ten inch rule, a quarter. That's plenty tight.
The top of the tension hoop should be the same height above the top of the head surface, all the way around. My hoop is about an 1/8" above the surface of the head when tightened to 90 on a drum dial.
I agree with the other posts about making sure your coord rod is set to neutral, then loosening the strings and all the hoop hooks and start from scratch on pulling down the head evenly, to the proper tension, so the hoop is an even height all the way around.
It isn't the height of the tension hoop above the head, it is that the tension hoop appears to be out of round.
I will go through jlsorbit's suggestions tomorrow - can't do it now as it is now evening in France and I have had a couple of glasses of wine ;). If that doesn't sort it, I will take some pictures and post here.
I agree with jlsorbit. Goodtime tension hoops are not super thick or rigid, so it could simply be a little bit less than round (no biggie) or it could be that someone has reefed down on the coordinator rod nuts one way or the other to try to adjust the action and egging the rim in the process. A little bit of that goes a long way. If the action needs adjustment, I'd look to shifting the neck up or down a little (usually a little play in the holes in the rim) or changing the bridge. Looking forward to hearing how this works out.
Ok, so I loosened the inside and outside nuts, and nothing changed. So I tightened the inside nut as much as I dared and left it for a while. And that didn't change anything.
But looking in detail at the pot assembly, I actually think it might be that the tension hoop isn't round. When I measure the underside of the pot, it is +/- 1 mm 25 cm side to side. Also the flange is flush with the pot all the way round. As is the head.
So it looks like the tension hoop is a bit less than round, and if the only potential problem this causes is that dirt will get in the gap between the head and the hoop, I will just ignore it and continue playing.
I can't see how tensioning the nuts will help because they are all about pulling down. Is that right? So I will leave them as they are for now.
On the brightside, I now know what goes on inside the pot, and my banjo has a couple of war wounds from some clumsy spanner work!
Good morning. Your photo didn't post, but it sounds like you have made the correct diagnosis. If you feel like wrenching on it some more, you can remove the hoop and probably gently bend it to be more round. Or you can play it. It's not hurting anything as long as it makes good contact with the flesh hoop on the head.
You are correct....the nuts only apply tension and have no bearing on roundness. And since you have determined that the pot is round, you want to end up with the nuts on the coordinator rod being neutral so they're not trying to make it otherwise.
It won't get any worse, and it looks like the banjo may have snuck through Deering's inspection with a slightly over-sized tension band. If you are the original owner you might shoot Deering a photo; they may send you a new, smaller tension band.