  In this paper, we conduct a comprehensive study of the Next-to-Minimal
Composite Higgs Model (NMCHM) extended with a dilaton field $\chi$ (denoted as
NMCHM$_\chi$). A pseudo-Nambu-Goldstone boson (pNGB) $\eta$, resulting from the
SO(6)$\to$SO(5) breaking, serves as a dark matter (DM) candidate. The inclusion
of the dilaton field is helpful for evading the stringent constraints from dark
matter direct detection, as it allows for an accidental cancellation between
the amplitudes of DM-nucleon scattering, an outcome of the mixing between the
dilaton and Higgs fields. The presence of the dilaton field also enriches the
phase transition patterns in the early universe. We identify two types of phase
transitions: (i) a 1-step phase transition, where the chiral symmetry and
electroweak symmetry breaking (EWSB) occur simultaneously, and (ii) a 2-step
phase transition, where the chiral symmetry breaking transition takes place
first, followed by a second phase transition corresponding to EWSB. Since the
first-order phase transitions can be strong due to supercooling in our model,
we also examine the stochastic background of gravitational waves generated by
these phase transitions. We find that these gravitational waves hold promise
for detection in future space-based gravitational wave experiments, such as
LISA, Taiji, BBO, and DECIGO.
