We study distributed user association in 5G and beyond millimeter-wave enabled heterogeneous networks using matching theory. We propose a novel and efficient distributed matching game, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">early acceptance</i> (EA), which allows users to apply for association with their ranked-preference base station in a distributed fashion and get accepted as soon as they are in the base station’s preference list with available quota. Several variants of the EA matching game with preference list updating and reapplying are compared with the original and stability-optimal deferred acceptance (DA) matching game, which implements a waiting list at each base station and delays user association until the game finishes. We show that matching stability needs not lead to optimal performance in other metrics such as throughput. Analysis and simulations show that compared to DA, the proposed EA matching games achieve higher network throughput while exhibiting a significantly faster association process. Furthermore, the EA games either playing once or multiple times can reach closely the network utility of a centralized user association while having much lower complexity.