These references may be more shallow than you desired, but they are both fun and lucid.
1) Noga Alon's Tools From Higher Algebra contains many things (or at least references to those things) that only require linear algebra at heart, such as Rayleigh's Principle.
2) A Course in Combinatorics by van Lint and Wilson is laced with gems in self-contained sections, such that each page is an adventure. You'll find Lots of techniques here that only require linear algebra, including the awkward-looking "interlacing property" of eigenvalues that have popped up way too much for me to ignore by now.
My favorite is actually the aforementioned Babai/Frankl manuscript, which is still very readable and useful. In theory you can still get it; in practice more difficult. First time I tried to order it I didn't get a reply at all.