  Valid hook configurations are combinatorial objects used to understand West's
stack-sorting map. We extend existing bijections corresponding valid hook
configurations to intervals in partial orders on Motzkin paths. To enumerate
valid hook configurations on $312$-avoiding permutations, we build off of an
existing bijection into a Motzkin poset and construct a bijection to certain
well-studied closed lattice walks in the first quadrant. We use existing
results about these lattice paths to show that valid hook configurations on
$312$-avoiding permutations are not counted by a $D$-finite generating
function, resolving a question of Defant's, and additionally to compute
asymptotics for the number of such configurations. We also extend a bijection
of Defant's to a correspondence between valid hook configurations on
$132$-avoiding permutations and intervals in the Motzkin-Tamari posets,
providing a more elegant proof of Defant's enumeration thereof. To investigate
this bijection, we present a number of lemmas about valid hook configurations
that are generally applicable and further study the bijections of Defant's.
