Square-free powers of square-free monomial ideals relate homological properties of monomial ideals with combinatorial matching structures. Let $ I $ be any square-free monomial ideal, and let $ \mathcal {H}_I $ denote its associated hypergraph. We refine the concept of $ k $-admissible matching, originally introduced by Erey and Hibi for graphs, by defining a generalized version for arbitrary hypergraphs. This new notion allows us to establish a sharp lower bound for the Castelnuovo–Mumford regularity of the $ k\text {th} $ square-free power $ I^{[k]} $. When $ I $ is equigenerated, the bound is given in terms of a new combinatorial invariant, the $ k $-admissible matching number $ \mathrm {aim}(\mathcal {H}_I, k) $, provided $ I^{[k]} \neq 0 $. Surprisingly, for edge ideals $ I(G) $ of graphs, this gives the first general lower bound on the regularity of $ I(G)^{[k]}$. When $ G $ is a forest, our invariant coincides with the $ k $-admissible matching number of Erey and Hibi. Moreover, we prove that for any block graph $ G $, the bound becomes an exact formula, thereby generalizing the known regularity formula for forests. We conclude by proposing a conjecture on the regularity of square-free powers of edge ideals of chordal graphs.