We design a deterministic algorithm for the $(1+ε)$-approximate maximum matching problem. Our primary result demonstrates that this problem can be solved in $O(ε^{-6})$ semi-streaming passes, improving upon the $O(ε^{-19})$ pass-complexity algorithm by [Fischer, Mitrović, and Uitto, STOC'22]. This contributes substantially toward resolving Open question 2 from [Assadi, SOSA'24]. Leveraging the framework introduced in [FMU'22], our algorithm achieves an analogous round complexity speed-up for computing a $(1+ε)$-approximate maximum matching in both the Massively Parallel Computation (MPC) and CONGEST models. The data structures maintained by our algorithm are formulated using blossom notation and represented through alternating trees. This approach enables a simplified correctness analysis by treating specific components as if operating on bipartite graphs, effectively circumventing certain technical intricacies present in prior work.