If I recall correctly, Stanley does not discuss the so-called "Tits monoid" associated to a hyperplane arrangement. The relationship between the Tits monoid and the lattice of flats of a hyperplane arrangement is the main focus of the book "Topics in hyperplane arrangements" by Aguiar and Mahajan (it's free online at http://pi.math.cornell.edu/~maguiar/surv-226.pdf). Actually, they have another more recent book on hyperplane arrangements as well - I'm not quite sure the difference between the two. Anyways, this is definitely an "algebraic" direction in hyperplane arrangement theory.
The study of free arrangements/the module of logarithmic derivations -- as championed by Terao -- is another completely different algebraic (or more specifically, algebro-geometric) direction in hypreplane arrangement theory. This one is arguably more connected (at least "morally") to the stuff in Stanley's notes (e.g. I believe Stanley mentions free arrangements, at least in an exercise). This is discussed e.g. in the book by Orlik and Terao mentioned in the comment above.