  We generalize the recently proposed Stepped Partially Acoustic Dark Matter
(SPartAcous) model by including additional massless degrees of freedom in the
dark radiation sector. We fit SPartAcous and its generalization against
cosmological precision data from the cosmic microwave background, baryon
acoustic oscillations, large-scale structure, supernovae type Ia, and Cepheid
variables. We find that SPartAcous significantly reduces the $H_0$ tension but
does not provide any meaningful improvement of the $S_8$ tension, while the
generalized model succeeds in addressing both tensions, and provides a better
fit than $\Lambda\mathrm{CDM}$ and other dark sector models proposed to address
the same tensions. In the generalized model, $H_0$ can be raised to
$71.4~\mathrm{km/s/Mpc}$ (the 95% upper limit) if the fitted data does not
include the direct measurement from the SH0ES collaboration, and to
$73.7~\mathrm{km/s/Mpc}$ (95% upper limit) if it does. A version of
$\texttt{CLASS}$ that has been modified to analyze this model is publicly
available at https://github.com/ManuelBuenAbad/class_spartacous
