Non-bijective conformal maps between annuli

If a bounded holomorphic function $f$ on the unit disk has boundary value zero on a positive measure subset, then $f\equiv 0$ (this is a well known fact from the theory of Hardy spaces, and it holds more generally).

This rules out the existence of functions such as the ones you describe above (by conformal mapping of a simply connected piece of your annulus).

Here is another way to get the answer: Any map as in your question is a covering map. Hence, if the degree is one, it is in fact a conformal isomorphism.

Note that this implies (as does Christian's answer) that the assumption of being degree $1$ is not necessary.

It also shows that, in general, the only locally conformal maps between annuli that map boundaries to boundaries are the obvious ones, with the corresponding relations between moduli.