Suppose $f: Y \rightarrow X$ is a covering map between compact Hausdorff spaces $X$ and $Y$. Then $f$ induces a algebra homomorphism $f^*:C(X) \rightarrow C(Y)$ and gives $C(Y)$ the structure of a finitely generated projective module over $C(X)$. Hence $C(Y)$ can be viewed as the sections of a complex vector bundle $E \rightarrow X$.

I am interested in the converse. Given a complex vector bundle $E \rightarrow X$, when is $E$ defined from a covering map as above? In particular, is this the case if the first Chern character $c_1(E) \in \check{H}^2(X ; \mathbb{Z})$ vanishes? Is there always a complex line bundle $L \rightarrow X$ such that $E \otimes L$ is defined by a covering map as above?