[1]:http://pages.bangor.ac.uk/topgpds.html

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][1],  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 $p$. 

The proofs in the book go by modelling covering maps of spaces by covering morphisms of groupoids. This modelling seems to me particularly convenient, as against the usual modelling in terms of actions of groups,  when considering lifting of maps.