Adapting Reiher's proof of Kemnitz's conjecture, we obtain two refinements of a theorem of Schmid and Zhuang. Our main results provide improved upper bounds for the Erdős-Ginzburg-Ziv constant of rank-two-like $p$-groups, and their direct products with cyclic groups of order coprime to $p$. In particular, we determine the exact value of this constant, and also confirm a conjecture of Gao, for a new infinite family of groups of arbitrarily large rank.