A search for a new Z′ gauge boson predicted by Lµ-Lτ models, based on charged-current Drell-Yan production, pp→W±(∗)→Z′μ±ν→μ±μ μ±ν, is presented. The data sample used corresponds to an integrated luminosity of 140 fb-1 of proton-proton collisions at s=13 TeV recorded by the ATLAS detector at the Large Hadron Collider. The search examines a final state of 3μ plus large missing transverse momentum. Upper limits are set on the Z′ production cross section times branching ratio in the mass range of 5-81 GeV. After combining with the previous Z′ search using the neutral-current Drell-Yan production with a 4μ final state, the most stringent exclusion limits to date are achieved in the parameter space of the Z′ coupling strength and mass.