  We consider a gauge-singlet complex scalar field $\Phi$ with a global $U(1)$
symmetry that is spontaneously broken at some high energy scale $f_a$. As a
result, the angular part of the $\Phi$-field becomes an axion-like particle
(ALP). We show that if the $\Phi$-field has a non-zero coupling $\kappa$ to the
Standard Model Higgs boson, there exists a certain region in the $\left(f_a,
\kappa\right)$ parameter space where the global $U(1)$ symmetry-breaking
induces a strongly first order phase transition, thereby producing stochastic
gravitational waves that are potentially observable in current and future
gravitational-wave detectors. In particular, we find that future
gravitational-wave experiments such as TianQin, BBO and Cosmic Explorer could
probe a broad range of the energy scale $10^3 \, {\rm GeV} \lesssim f_a
\lesssim 10^{8} \, {\rm GeV}$, independent of the ALP mass. Since all the ALP
couplings to the Standard Model particles are proportional to inverse powers of
the energy scale $f_a$ (up to model-dependent ${\cal O}(1)$ coefficients), the
gravitational-wave detection prospects are largely complementary to the current
laboratory, astrophysical and cosmological probes of the ALP scenarios.
