I was trying to do this exercise from Hatcher's book:
Let $p:F\to E,q:E\to X$ maps such that $q$ and $q\circ p$ are covering maps. If $X$ is locally path-connected, then show that $p$ is a covering map.
I tried it as follows:
Let $e\in E$. I must find an evenly covered neighborhood (with respect to $p$) of $e$. I think I can't do this just now, but I can take a neighborhood $U$ of $q(e)$ which is path-connected and evenly covered with respect to both $q$ and $q\circ p$. Then $$q^{-1}(U)=\bigsqcup_{i\in I} V_i, (q\circ p)^{-1}(U)=\bigsqcup_{j\in J}W_j$$ such that $q|_{V_i},(q\circ p)|_{W_j}$ are homeomorphisms for each $i,j$. Fix $i$ such that $e\in V:=V_i$.
I want to prove that $p^{-1}(V)$ is the disjoint union of a subfamily of the $W_j$'s, each of them mapped homeomorphically by $p$ onto $V$. But I don't know how to get there.