The usual classification of covering spaces (stated in terms of a categorical equivalence) requires the conditions "locally path connected" and "semi-locally simply connected."
Theorem 1: If $X$ is a path-connected, locally path-connected and semi-locally simply connected space, then the category of covering spaces over $X$ is equivalent to the category of left $\pi_1(X)$ actions on sets.
If $X$ is locally path connected, the fiber bundles with totally path disconnected fibers are actually in bijective correspondence with the (genuine) coverings of $X$, that is, any such fiber bundle corresponds to a unique covering map. To see this we can exploit the fact that these fiber bundles cannot possibly be locally path connected unless they are already covering maps.
Given any space $Y$, let $lpc(Y)$ be the underlying set of $Y$ with the topology generated by the path components of open sets of $Y$. This topology is finer than the topology on $Y$ and the continuous identity function $c_Y:lpc(Y)\to Y$ is universal in the sense that if $Z$ is locally path connected and $f:Z\to X$ is continuous, then so is $f:Z\to lpc(X)$. In other words, $lpc$ is left adjoint to the inclusion of locally path connected spaces in to all topological spaces and $c_Y$ is the counit.
Since $Z=[0,1]^n$ is locally path connected, $lpc(X)$ is path connected whenever $X$ is (since both topologies admit exactly the same paths) and $lpc(X)\to X$ induces an isomorphism $\pi_1(lpc(X),x)\to \pi_1(X,x)$ for any $x$.
Now if $p:E\to X$ is a fiber bundle with totally disconnected fiber $F$, then there is a unique covering map $p':lpc(E)\to X$ such that $p'=p\circ c_{E}$. If we forget the topology on $F$, then the $\pi_1(X)$-action from the bundle gives the $\pi_1(X)$-action that characterizes the covering $p'$.
So if $X$ is path connected and locally path connected, the category of coverings over $X$ and the category of fiber bundles over $X$ with totally path-disconnected fiber are equivalent. They must "embed" into the category of $\pi_1(X)$-sets, however, neither must be equivalent to the category of $\pi_1(X)$-sets unless $X$ is semilocally-simply connected.
Some examples of generalizations of covering space theory
I will do my best to address your question about the other generalizations of covering space theory, however, there are many of them I won't be able to get to. I would suggest looking at some of the reference in the papers I'll mention below. I should clarify with this: There is no "correct" generalization of covering space theory. The usefulness of a given generalization depends on the intended application and these can vary greatly.
1) Covering maps: There are ways to classify covering maps (in the usual sense) of arbitrary locally path connected spaces. One is to use shape theoretic/steenrod homotopy methods for finite sheeted coverings or one can use topologized versions of the fundamental group to classify arbitrary coverings. For instance in 2 and [3] it is shown that open subgroups of $\pi_(X)$ (with a certain group topology) which contain an open normal subgroup classify the coverings of $X$ (even when $X$ is not semilocally simply connected). The open subgroups in general classify semicoverings (see below).
For finite sheeted coverings see:
1 L.J. Hernández-Paricio, V. Matijevic, Fundamental groups and finite sheeted coverings, J. Pure Appl. Algebra 214 (2010), no. 3, 281–296.
For arbitrary coverings see:
2 H. Fischer, A. Zastrow, A core-free semicovering of the Hawaiian Earring, Topology Appl. 160 (2013), no. 14, 1957–1967.
[3] J. Brazas, Semicoverings, coverings, overlays and open subgroups of the quasitopological fundamental group, Topology Proc. 44 (2014) 285-313.
2) Overlays: One of the most recognized generalizations of covering space theory is R.H. Fox's theory of overlays. Overlays are "nice" covering maps where you can uniquely lift chains of neighborhoods in a trivializing cover. Overlays do not provide more information about $\pi_1$ but the reason they are so important is that they admit a convenient classification for all compact metric spaces in terms of actions of symmetric groups on factors of the fundamental pro-group.
[4] Ralph H. Fox, On shape, Fund. Math. 74 (1972), no. 1, 47–71.
[5] Sibe Mardešic and Vlasta Matijevic, Classifying overlay structures of topological
spaces, Topology Appl. 113 (2001), no. 1-3, 167–209.
3) Semicoverings: Ok, so I am a little biased on this one, but semicovering admit a categorical classification just like the one for coverings. Semicoverings are much more general than coverings and the classification applies to all locally path connected - and even some non-locally path connected - spaces. Here $\pi_1(X)$ has a certain topology and the category of semicoverings over $X$ is equivalent to the category of continuous $\pi_1(X)$-actions on discrete spaces. In other words, open stabilizer subgroups classify the semicoverings. The upshot is that this has led to solutions to long-standing open subgroup problems in the theory of topological groups [7].
[6] J. Brazas, Semicoverings: A generalization of covering space theory, Homology Homotopy Appl. 14 (2012), no. 1, 33–63.
[7] J. Brazas, Open subgroups of free topological groups, To appear in Fund. Math. 2014 http://arxiv.org/abs/1209.5486.
4) Generalized coverings defined only in terms of unique lifting properties: Introduced by Hanspeter Fischer and Andreas Zastrow for locally path connected spaces in their very insightful paper [8], these have been used more recently to understand the algebraic structure of fundamental groups of wild spaces. These behave just like covering maps in nearly every respect except local triviality (or even being a local homeomorphism). There does not seem to be any known categorical classification of these due to the fact that the general occurrence of unique path lifting property is difficult to characterize.
[8] H. Fischer, A. Zastrow, Generalized universal covering spaces and the shape group, Fund. Math. 197 (2007), 167–196.
[9] J. Dydak, Coverings and fundamental groups: a new approach, 2011 http://arxiv.org/abs/1108.3253.
Other approaches include studying coverings of uniform spaces and their generalizations, attaching data to locally constant presheaves, etc. If anyone is interested in more, I suggest looking at the references in the papers mentioned above to see what has been done. For instance, I try to give a general overview in the introduction of [3].
