A search for the dimuon decay of the Standard Model Higgs boson, $H\rightarrow\mu\mu$, is presented using $pp$ collision data at $\sqrt{s}=13.6~\mathrm{TeV}$ collected with the ATLAS experiment during 2022–2024, corresponding to an integrated luminosity of $165~\mathrm{fb}^{-1}$. The $H\rightarrow\mu\mu$ channel provides a direct
probe of the Higgs-boson coupling to second-generation charged leptons, but the search is experimentally challenging because of its small branching fraction and the large Drell-Yan background. Events are classified into 23 mutually exclusive categories targeting ggF, VBF, $VH$, and $t\bar{t}H$ production. Multivariate techniques are used to enhance the signal sensitivity, and the signal is extracted from simultaneous fits to the dimuon invariant-mass spectra in each category. The result is combined with the previous ATLAS Run 2 result. The combined analysis gives a signal strength of $\mu=1.4\pm0.4$ and an observed (expected) significance of $3.4\sigma$ ($2.5\sigma$), providing evidence for the $H\rightarrow\mu\mu$ decay with the ATLAS detector.