… Does it detach from the Magic Ring, or is it left on the table, possibly re-attaching to a different Keeper?

A: The Magic Ring, while face-out in your hand, is still considered to be in a state of Having-Already-Been-Played, and the Evil can stay attached. In this case, the Evil should be held face-out in your hand, because it is attached to the Magic Ring.

If, however, you deactivate the Magic Ring by turning it back face-in with the rest of your hand, NOW it is fully in your hand, and Evil must detach and go back onto the table, attaching to a different Keeper if one is available. This would be a perfectly valid way to detach the Evil from the Magic Ring.