Is a covering space of a manifold always a manifold

Assume $M$ is a manifold and $q : E \to M$ is a covering map. I have been told a few times that a covering space of a manifold is again a manifold. Indeed, it is easy to verify that $E$ is both Hausdorff and locally euclidean. I am worried about whether $E$ needs to be second countable.

If $E$ is not connected, then it does not need to be second countable. Take the standard covering map $\coprod_{i \in \mathbb{R}} M \to M$.

Question: If $E$ is connected, then why is it second countable?

Because of Poincaré-Volterra's theorem.
The best reference is, as usual, Forster's great Lectures on Riemann surfaces , Lemma 23.2, page 186.
A less self-contained but more general version (surprise, surprise...) is to be found in Bourbaki's General Topology : it is the very last result of Chapter 1.

In the case that the manifold has a PL structure, I believe there's an elementary proof as follows.

Since $E$ is connected, the degree of the covering is the size of the fundamental group of $M$.

Also, the fundamental group of $M$ must be countable, as follows. $M$ is homeomorphic to a locally finite simplicial complex. The fundamental group comes from paths in the 1-skeleton of that complex, considered as a graph. Since the simplicial complex is locally finite, there are only countably many such paths, and therefore the fundamental group is countable.

So the degree of the covering is countable, and we can lift a countable basis of $M$ to get a countable basis of $E$. Therefore $E$ is second countable.