Inconsistencies between the predictions based on the \(\varLambda \textrm{CDM}\) model and astrophysical observations at dwarf and galaxy scales motivate alternatives to WIMPs. A compelling alternative is the self-interacting dark matter (SIDM), which not only addresses the anomalies on small scales but also aligns seamlessly with the large-scale \(\varLambda \textrm{CDM}\) predictions. Nevertheless, the thermal relic of SIDM tends to be under-abundant within the standard cosmological framework, which considers dark matter decoupling from the cosmic soup in a radiation-dominated epoch. To tackle this challenge, our approach involves introducing a non-standard expansion history of the universe by incorporating a non-radiation-like component into the cosmic soup. In this modified scenario, SIDM decouples at an early epoch compared to the standard model, resulting in an enhanced relic density that matches observed values. We consider our SIDM candidate to be a pseudo-Dirac fermion to nullify its diagonal vector interaction, thereby circumventing direct search constraints. Through this, we identify a parameter space that remains consistent with astrophysical observations and relic density criteria.