For each surface $S$ of genus $g>2$ we construct pairs of conjugate pseudo-Anosov maps, $\varphi_1$ and $\varphi_2$, and two non-equivalent covers $p_i\colon \tilde S \longrightarrow S$, $i=1,2$, so that the lift of $\varphi_1$ to~$\tilde S$ with respect to $p_1$ coincides with one of $\varphi_2$ with respect to $p_2$.
The mapping tori of the $\varphi_i$ and their lift provide examples of pairs of hyperbolic $3$-manifolds so that the first is covered by the second in two different ways.
"A Note on Covers of Fibred Hyperbolic Manifolds." Publ. Mat. 61 (2) 517 - 527, 2017. https://doi.org/10.5565/PUBLMAT6121707