I didn't say that well. I mean, non-zero everywhere. There can be bumps with most of the support, but integrating |psi|^2 over any region gives a non-zero result, if mostly below the order of, say 10^-100 outside of your two bumps. That is, from a probabilistic point of view, it can be discounted without a thought, but from a "this is a real world" it does not seem to me to be so easy to discard.

That is to say, where is the magnitude of the wave function being used in this interpretation?