I'm not a big fan of Peter May's books, so I'm not sure if the original poster will like these
UPDATED.In particular After Peter May and Kate Ponto released their new book, none of them there are too married very readable introductions to category theorymany of the topics on the "second level" of algebraic topology.However,
There is a wonderful book on Cohomology Operations by Mosher and Tangora. It is thin (and only discusses one topic), but very nice.
May & Ponto's new book is very nice. It covers three topics (Professor May's comment above has details) + an appendix on spectral sequences, which is short but very much to the point. I used to fear that any book by May was secretly about category theory, but that is not true about 3/4 of this one (unless the secret is hidden too well).
There is a pretty good, and more comprehensive book by Fomenko and Fuks (or Fuchs?) on homotopy theory. I've only seen the Russian version (so I can't vouch for the translation). It's also not very well-known, and not very easy to find, which is a shame (the Russian version is more obtainable). It has a lot of stuff, including one of the nicer introductions to spectral sequences (although I don't know a single book that does this well. Serre's thesis is nice, Hatcher's notes are OK, but this seems to be a topic best learned in a good class). It's also very readable. Here's a review (institutional access probably required) with a description of its contents.
