Let $X$ be a smooth finite type separated connected Deligne-Mumford stack over $\mathbb C$.

Does there exist a finite etale morphism $Y\to X$ with $Y$ a scheme?

What if $X$ is an algebraic space (i.e., trivial stabilizers)?

Edit: I changed the old question to a different question which should be more clear. An answer to the new question would help a lot in answering the old question.