A search for the dimuon decay of the Standard Model (SM) Higgs boson is performed using data corresponding to an integrated luminosity of 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ collected with the ATLAS detector in Run2 $pp$ collisions at $s=13$ TeV at the Large Hadron Collider. The observed (expected) significance over the background-only hypothesis for a Higgs boson with a mass of 125.09 GeV is $2.0\sigma$ ($1.7\sigma$). The observed upper limit on the cross section times branching ratio for $pp\to H\to \mu \mu$ is 2.2 times the SM prediction at 95% confidence level, while the expected limit on a $H\to \mu \mu$ signal assuming the absence (presence) of a SM signal is $1.1$ ($2.0$). The best-fit value of the signal strength parameter, defined as the ratio of the observed signal yield to the one expected in the SM, is $\mu =1.2\pm 0.6$.