Can a stable matching that achieves high equity among the two sides of a market be reached in quadratic time? The Deferred Acceptance (DA) algorithm finds a stable matching that is biased in favor of one side; optimizing apt equity measures is strongly NP-hard. A proposed approximation algorithm offers a guarantee only with respect to the DA solutions. Recent work introduced Deferred Acceptance with Compensation Chains (DACC), a class of algorithms that can reach any stable matching in O(n^4) time, but did not propose a way to achieve good equity. In this paper, we propose an alternative that is computationally simpler and achieves high equity too. We introduce Monotonic Deferred Acceptance (MDA), a class of algorithms that progresses monotonically towards a stable matching; we couple MDA with a mechanism we call Strongly Deferred Acceptance (SDA), to build an algorithm that reaches an equitable stable matching in quadratic time; we amend this algorithm with a few low-cost local search steps to what we call Deferred Local Search (DLS), and demonstrate experimentally that it outperforms previous solutions in terms of equity measures and matches the most efficient ones in runtime.