After a quick play I see no reason to stop it from working depending on the finger you wear it in. And the other rings would preferably not be the same colour as Odyssey.
The only down side I see would be that too many rings may detract from the effect because the spectator would be seeing more than one ring in play at a time.
You can easily wear Odyssey with other rings. I can see wearing rings and removing them to emphasize that you're wearing a real ring being a benefit if that's what you're into when approaching the trick.
I agree, wearing other rings may bring up the question of why you're not able to do the same things with others. Best to make them think it's a normal ring and keep the focus on your amazing skills. regards RJ