Let $N$ be an oriented hyperbolic 3-manifold of finite volume and let $\Delta \subset N$ be a smooth connected compact subdomain such that the restriction of the injectivity radius function of $N$ to $\Delta$ is larger than some $\varepsilon > 0$ ($\Delta$ could be a thick part of $N$). Is there some finite cover $\Pi\colon \widehat{N} \to N$ such that the injectivity radius function of $\widehat{N}$ restricted to any component of $\Pi^{-1}(\Delta)$ is greater than $\varepsilon + 1$?