  We investigate the temperature-dependent production of feebly interacting
massive dark matter particle (FIMP DM) within a $Z_2$ model, incorporating two
$Z_2$-odd scalar fields. In specific parameter regions, three distinct
mechanisms individually dominate the production of the FIMP DM. These
mechanisms include the semi-production process, commonly known as the
``exponential growth'' process, the production from pair annihilations of the
bath particles, and the three-body decay process. It is crucial to consider the
thermal history during the evolution of FIMPs, as it involves multiple phase
transitions occurring prior to the freeze-in of dark matter. Consequently, the
scalar masses experience thermal variations, leading to a distinctive
evolutionary trajectory for FIMPs when compared to scenarios without accounting
for the thermal effects. Notably, the unique pattern of FIMP evolution is
accompanied by the production of gravitational waves, presenting promising
opportunities for detection using forthcoming interferometers.
