  The Hubble tension persists as a challenge in cosmology. Even early dark
energy (EDE) models, initially considered the most promising for alleviating
the Hubble tension, fall short of addressing the issue without exacerbating
other tensions, such as the $S_8$ tension. Considering that a negative dark
matter (DM) equation of state (EoS) parameter is conducive to reduce the value
of $\sigma_8$ parameter, we extend the axion-like EDE model in this paper by
replacing the cold dark matter (CDM) with DM characterized by a constant EoS
$w_{\rm dm}$ (referred as WDM hereafter). We then impose constraints on this
axion-like EDE extension model, along with three other models: the axion-like
EDE model, $\Lambda$WDM, and $\Lambda$CDM. These constraints are derived from a
comprehensive analysis incorporating data from the Planck 2018 cosmic microwave
background (CMB), baryon acoustic oscillations (BAO), the Pantheon compilation,
as well as a prior on $H_0$ (i.e., $H_0=73.04\pm1.04$, based on the latest
local measurement by Riess et al.) and a Gaussianized prior on $S_8$ (i.e.,
$S_8=0.766\pm0.017$, determined through the joint analysis of
KID1000+BOSS+2dLenS). We find that although the new model maintains the ability
to alleviate the Hubble tension to $\sim$ 1.4$\sigma$, it still exacerbate the
$S_8$ tension to a level similar to that of the axion-like EDE model.
