What to read alongside Hatcher's Algebraic Topology

It's my experience that one doesn't usually 'read through Hatcher' at least not in the traditional sense of reading a book cover to cover. The material in Hatcher is simply too broad, ranging from basic introduction to algebraic topology all the way to very abstract homotopy theory. In my experience it's more of a reference book - after you've read the first chapter or two, you pick the right bits to read when you need them, often not in the order they're written.

The question in your last paragraph is more manageable as you're asking for extra material on the homological algebra covered in the text. My suggestion would be any of the classical texts on homological algebra such as Cartan-Eilenberg for a classical reference or any of a good range of modern texts which may be gentler introductions. There's a helpful thread here which covers books which might be of use.

I think if you are looking for complementary texts for other parts of Hatcher, you would get a better response if you made a separate question for each specific part. For instance "I'm looking for a more in-depth introduction to similar material on the surface theory covered by Hatcher - especially with regard to covering spaces." or "Hatcher glosses over some of the details in the section on Postnikov towers. Is there a reference with more details and possibly some examples?"

I would recommend Weibel's "Introduction to Homological Algebra", just to reach a good level on the algebraic side following a rather modern treatment. As far as topology is concerned, after having read Hatcher a good step could be May's "Concise course in Algebraic Topology": it is far more "unified" than Hatcher and it provides a strong homotopical foundation to the subject. Finally, you can consider also to pass to the abstract side, of course after having understood the fundamental notions: I am thinking of abstract homotopy theory, which would give rise to a whole new post by itself!

I like Topology and Geometry by Glen E. Bredon, especially as a reference for cohomology and homology products.

I second Weibel's book on homological algebra, especially for the universal coefficient theorem.