  A phase shift in the acoustic oscillations of cosmic microwave background
(CMB) spectra is a characteristic signature for the presence of non-photon
radiation propagating differently from photons, even when the radiation couples
to the Standard Model particles solely gravitationally. It is well-established
that compared to the presence of free-streaming radiation, CMB spectra shift to
higher $\ell$-modes in the presence of self-interacting non-photon radiation
such as neutrinos and dark radiation. In this study, we further demonstrate
that the scattering of non-photon radiation with dark matter can further
amplify this phase shift. We show that when the energy density of the
interacting radiation surpasses that of interacting dark matter around
matter-radiation equality, the phase shift enhancement is proportional to the
interacting dark matter abundance and remains insensitive to the radiation
energy density. Given the presence of dark matter-radiation interaction, this
additional phase shift emerges as a generic signature of models featuring an
interacting dark sector or neutrino-dark matter scattering. Using neutrino-dark
matter scattering as an example, we numerically calculate the amplified phase
shift and offer an analytical interpretation of the result by modeling photon
and neutrino perturbations with coupled harmonic oscillators. This framework
also explains the phase shift contrast between self-interacting and
free-streaming neutrinos. Fitting models with neutrino-dark matter or dark
radiation-dark matter interactions to CMB and large-scale structure data, we
validate the presence of the enhanced phase shift, affirmed by the linear
dependence observed between the preferred regions of the sound horizon angle
$\theta_s$ and interacting dark matter abundance.
