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 theH0tension but does not provide any meaningful improvement of theS8tension, while the generalized model succeeds in addressing both tensions, and provides a better fit than ΛCDM and other dark sector models proposed to address the same tensions. In the generalized model,H0can be raised to 71.4 km/s/Mpc (the 95% upper limit), reducing the tension, if the fitted data does not include the direct measurement from the SH0ES collaboration, and to 73.7 km/s/Mpc (95% upper limit) if it does. A version ofCLASSthat has been modified to analyze this model is publicly available athttps://github.com/ManuelBuenAbad/class_spartacous.