Reddit mentions of Putnam and Beyond

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Putnam and Beyond is exactly what you're looking for. The difficulty of the problems in each chapter (including proofs) start at, I would say, "easy to tricky", depending on your strength, and quickly ascend to "Putnam level" and, as the name suggests, "Beyond Putnam level".

The first chapters focus on specific methods of proof. Then the rest of the chapters focus on different areas of undergrad-level math. Each chapter has a few examples and useful tips, but it's mostly just problems. The second half of the book offers detailed solutions to each problem. At 814 pages, this book is bound to keep you entertained for a long time.
Proofs from The Book is not quite what you're looking for, but is instead a collection of some of the most elegant or well known proofs in mathematics (mostly discrete maths) that are accessible to an undergrad. There are, for example, 6 different proofs to the infinity of primes. Also highly entertaining and is helping me a lot in seeing proofs in a different light, appreciating different styles and approaches.

Tbh, I can't tell you from personal experience because after I got into university I stopped preparing for competitions and use my free time to study some other math instead.

Anyway, here are some books that I've heard are good for Putnam-esque preparation.

Putnam And Beyond

Problem-Solving Through Problems

Problems From The Book

Straight From The Book