  We propose a novel and comprehensive particle physics framework that
addresses multiple cosmological tensions observed in recent measurements of the
Hubble parameter, $S_8$, and Lyman-$\alpha$ forest data. Our model, termed
`{\bf SIDR+$\boldsymbol{z_t}$}' (Self Interacting Dark Radiation with
transition redshift), is based on an inelastic dark matter (IDM) scenario
coupled with dark radiation, governed by a $U(1)_D$ gauge symmetry. This
framework naturally incorporates cold dark matter (DM), SIDR, and the
interactions between these components. The fluid-like behavior of the dark
radiation component, effectively mitigates both the Hubble and $S_8$ tensions
by suppressing free-streaming effects. Simultaneously, the interacting DM-DR
system attenuates the matter power spectrum at small scales, potentially
reconciling discrepancies in Lyman-$\alpha$ (Ly-$\alpha$) observations. The
inelastic nature of DM provides a distinct temperature dependence for the DM-DR
interaction rate determined by the mass-splitting between the inelastic dark
fermions which is crucial for resolving the Ly-$\alpha$ discrepancies. We
present a cosmologically consistent analysis of the model by solving the
relevant Boltzmann equations to obtain the energy density and number density
evolution of different species of the model. The DR undergoes two ``steps" of
increased energy density when the heavier dark species freeze out and become
non-relativistic, transferring their entropy to the dark radiation and
enhancing $\Delta N{\rm eff}$. The analysis showcases the model's potential to
uphold the Big Bang Nucleosynthesis (BBN) prediction of $\Delta N_{\rm eff}$
but dominantly producing additional contributions prior to recombination, while
simultaneously achieving correct relic density of DM though an hybrid of
freeze-in and non-thermal production.
