A few years ago I too have been trying to look for that springy piece of.. whatever it is. but failed.
So what I've been using so far is a little improvement of the copy paper + sticky tape method.
I take pieces of leftover paper (normally the same type as the roll i'm planning on wrapping it with, so the PH, etc.. doesnt get messed up) and put velcro tapes on them as you can see from the picture.
It's 100% secure, doesnt damage the roll in any way, and costs next to nothing.