How were you practicing for the AMC? Studying specific topics or taking old tests and reviewing solutions? While the latter is a great way to actually prep for the tests, I would think a math club would focus more on a topic by topic basis, examining related concepts, theorems, etc.
My D’s math team is university-based and covers a large area, but I would think a similar approach would work for a HS team. They break up the year into 4 segments - Algebra, Geometry, Number Theory, Counting/Combinatorics, just like the AMC mostly does.
Fortunately, they publish much of their material, going back several years. Much of it is warm-up problems, challenge problems, etc., but some have definitions, theorems, proofs, etc. Much is also quite unstructured, going back many years, with different leaders, approaches. But it’s raw material that can at least show how others have done it and I suspect provide enough materials for a couple years of a HS club, as long as someone is willing to sort through it.
https://sites.google.com/site/westernpaarmlteam/downloads
http://www.math.cmu.edu/~mlavrov/arml/spring-2017.html (with links to previous years)
http://math.cmu.edu/~cargue/arml/archive.html#s17
In addition to teaching math outside the normal school curriculum, sessions on math history and key figures (Euler, Gauss, Ramanujn, Euclid) could be interesting. Have someone pick a person and spend a meeting discussing his/her life.
Maybe a topic on famous mathematical concepts and how they affected the world. I wore this just yesterday - https://www.amazon.com/Math-Seventeen-Equations-Changed-T-Shirt/dp/B082GGXDHS