Oops, what I originally wrote assumed U=X; didn't read the question properly.  Here's a fix:

First, suppose $L$ is simple.  $L$ is trivialied by some finite etale map $\tilde U\to U$ (this is what it means to have finite monodromy); then applying Grauert-Remmert (or SGA I, as mentioned in Georges Elencwajg's answer to [this question][1]), we may extend the map to a (possibly ramified) map $\pi: \tilde X\to X$. That is, $\pi^*L|_{\tilde U}=\underline{\mathbb{C}}^n$.  Then $\operatorname{Hom}|_U(L, \pi_*(\underline{\mathbb{C}}^n)|_{U})\simeq\operatorname{Hom}_{\tilde U}(\pi^*L|_{\tilde U}, \underline{\mathbb{C}}^n)$ is non-zero, so $L$ is a factor of $\pi_*\underline{\mathbb{C}}^n$ (using simplicity). Again, as $L$ is simple, it is in fact a factor of $\pi_*\underline{\mathbb{C}}$.  

Now if $L$ is not simple, decompose it into simple factors $L=\oplus L_i$, and let $\pi: {\tilde X_i}\to X$ be covers trivializing the $L_i$.  Setting $\tilde X=\bigsqcup {\tilde X_i}$ does the trick.


  [1]: http://mathoverflow.net/questions/60641/in-what-sense-is-the-etale-topology-equivalent-to-the-euclidean-topology