One of these cast iron models would be great. It has 5 folding arms with a total of 15 hooks and they all fold back against the wall... but alas they are very pricy. If money were no object this would be my choice.
Other things I found include an under the shelf model which pulls out....great for someone using a closet for storage. Another also goes under a shelf and swivels around... And there were many that you got only hooks but a shelf as a bonus...
I have a lot of vintage coat hooks in the barn and could have done something like this without too much effort..
But in the end it was the usual make-do with what I got. One of my problems was the center of the door was recessed and the rack wasn't stable which added to the flimsiness. So I added a brace (arrow) to the back which not only fill the recess gap, it stabilized the whole rack. The wooden pieces to hang the cups on were attached with nails and I replaced the nails with screws which also helped greatly. So this now my permanent home for my thread rings. Now I just have to get the rest of the thread on the rings...no small job in itself.