Assume that $X$ is a path-connected, locally path-connected, and semi locally simply-connected space. Then there is a way to know all path-connected covering spaces of $X$ by looking at the subgroups (and their conjugates) of $\pi_1(X)$. And we can combine different covering spaces to generate new (non-connected) covering spaces.
Can we generate all covering spaces of $X$ in this way?
Yes. Doing it this way allows for a cleaner statement of the Galois correspondence, which naturally generalizes even to the case that $X$ is not assumed to be path-connected:
The connected covering spaces correspond to the transitive $\pi_1(X)$-sets, which correspond to conjugacy classes of subgroups. But this way of stating the Galois correspondence tells you not only what non-connected covering spaces look like but also what maps between covering spaces look like, in a nice clean way. The resulting category is also much more nicely behaved than if we restrict to the connected case, e.g. it now has products and coproducts.
To generalize to the case that $X$ is not path-connected we replace the fundamental group $\pi_1(X)$ with the fundamental groupoid $\Pi_1(X)$.