  We argue that a large region of so far unconstrained parameter space for
axion-like particles (ALPs), where their couplings to the Standard Model are of
order $(0.01\!-\!1)\,\mbox{TeV}^{-1}$, can be explored by searches for the
exotic Higgs decays $h\to Za$ and $h\to aa$ 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
$a\to\gamma\gamma$, even if the relevant couplings are loop suppressed and the
$a\to\gamma\gamma$ branching ratio is less than~1.
