After years of catching glimpses of peoples' ring inventories, via screenies, I've noticed that almost everyone organizes their rings differently. I, for example, like to keep the top row empty.
That way when I get new ring drops, I know what they are and don't have to search for them or wonder where my old drops ended and the new ones begin. It's all right there at the top. Second row is comprised of the rings I'm most likely to swap out with the ones in my tray, depending on what I'm planning to do. Then the rest...in no particular order...more or less. Then my extras.
After seeing what new ring I have, I usually take a few minutes to organize them into my inventory so the top row is empty again for my next run.