Jauch, Piron, Ludwig -> QFT?

I think you are looking for things in the line of

• R Haag Local Quantum Physics: Fields, Particles, Algebras
• R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That

are you? Still, Zee's is a good idea, and some 1960 book to bridge the gap (EDIT: I was thinking some practical "lets calculate" book, as Bjorken Drell volumes, but from the point of view of fundamentals, also 1965 Jost The general theory of quantized fields could be of interest for the OP).

There are three QFT books that I would recommend.

Anthony Zee's Quantum Field Theory in a Nutshell is a very good book written for advanced undergraduates/beginning grad students. It is much more physics-y than math-y in focus, with everything motivated as carefully as possible. While the book is definitely quantitative, he puts a bigger priority on learning the conceptual underpinnings and the why of QFT than he does on either mathematical rigor or heavy calculation.

Peskin and Schroder's An Introduction to Quantum Feild Theory is a bit more advanced than Zee's book, and much less physically motivated. There is a much larger focus on calculating cross-sections in this book, though. It's also a bit older, so some newer developments aren't discussed. It's very clearly written for those more interested in phenomonlogy than fundamental theory, too. But you'll finish this book knowing how to do QFT, certainly.

And then, there's Stephen Weinberg's three part quantum field theory series. These books are quite mathematically rigorous, and also quite difficult to read. You'll understand the underlying math if you start with this book (and it sounds like that's what you want from your question and comment), but teaching yourself QFT out of this at a first pass would be like trying to learn E&M out of Jackson before having heard of an electric field. There is a lot of insight and careful mathematical development that you can only find in these books, though.