[1]:http://groupoids.org.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 $Y$; 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 above 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.