First, suppose $L$ is simple. Let $\pi: \tilde X\to X$ be a finite cover trivializing $L$ (this is what it means to have finite monodromy); that is, $\pi^*L=\underline{\mathbb{C}}^n$. Then $\operatorname{Hom}(L, \pi_*\underline{\mathbb{C}}^n)\simeq\operatorname{Hom}(\pi^*L, \underline{\mathbb{C}}^n)$ 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.