We argue that a large region of so-far unconstrained parameter space for axionlike particles (ALPs), where their couplings to the standard model are of order (0.01-1) TeV^{-1}, can be explored by searches for the exotic Higgs decays h→Za and h→aa in run 2 of the LHC. Almost the complete region in which ALPs can explain the anomalous magnetic moment of the muon can be probed by searches for these decays with subsequent decay a→γγ, even if the relevant couplings are loop suppressed and the a→γγ branching ratio is less than 1.