  Axion-like particles (ALPs) are a compelling candidate for dark matter (DM),
whose production is associated with the formation of a string-wall network. If
walls bounded by strings persist, which requires the potential to have multiple
local minima ($N>1$), they must annihilate before they become dominant. They
annihilate mostly into gravitational waves and non-relativistic ALPs. We show
that for ALPs other than the QCD axion these gravitational waves, if produced
at temperatures below 100 eV, could be detected by future cosmological probes
for ALPs with mass from $10^{-16}$ to $10^{6}$ eV that could constitute the
entirety of the DM.
