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I feel it would be difficult to deal with non locally path connected spaces and covering spaces, though I may be wrong. However the existence of a covering map $f: X \to Y$ implies certain local conditions on $Z$; these are usually stated in the case $f$ is a universal cover, but the more general case is in Chapter 10 of Topology and Groupoids, Section 5, as that $Y$ is semilocally $\chi_f$-connected, which means that each point $y \in Y$ has a neighbourhood $U$ such that for all $x \in X$ with $f(x)=y$ the image of $\pi_1(U,y)$ in $\pi_1(Y,y)$ is contained in the image under $f$ of $\pi_1(X,x)$. So to get a lift of $p: Z \to Y$ to all possible base points you seem to need a similar condition on $Z$ and $p$.