  An axion-like-particle (ALP) in the post-inflationary scenario with domain
wall number $N>1$ can be dark matter if the residual $\mathbb{Z}_N$ symmetry
has a small explicit breaking. Although we cannot determine the full dynamics
of the system reliably, we provide evidence that such an ALP can account for
the observed dark matter abundance while having a relatively small decay
constant and consequently a possibly large coupling to photons. In particular,
we determine the number of domain walls per Hubble patch around the time when
they form using numerical simulations and combine this with analytic
expectations about the subsequent dynamics. We show that the strongest
constraint on the decay constant is likely to come from the dark matter ALPs
being produced with large isocurvature fluctuations at small spatial scales. We
also comment on the uncertainties on the dark matter small-scale structure that
might form from these overdensities, in particular pointing out the importance
of quantum pressure in the $N=1$ case.
