This study addresses the challenging trade-off between maximizing environmental benefits and ensuring social fairness in a domestic carbon tax context. Focusing on France, where the pathway to decarbonization has faced public opposition, we explore four redistribution policies to reverse the regressive effect of a euro100/t of CO2e carbon tax. While the homogeneous redistribution scheme reaches progressivity, its social and environmental implications are not without issues. A substantial portion of the emissions reduction achieved by the carbon tax dissipates due to increased emissions from overcompensated households. Additionally, the compensation generates higher tax burden inequality within and between income groups. Three tailored scenarios are designed to overcome misguided redistribution and maintain the social objective: a vertical scenario targeting financially constrained households, a horizontal scenario focusing on vulnerable households, and a mixed scenario considering both aspects. The vertical scenario efficiently reduces tax burden dispersion among the most affected households, limiting post-redistribution emissions (a mere 0.65% increase) and proving cost-effective for the government. While the horizontal scenario requires a larger share of tax revenue, it successfully reduces overall tax burden dispersion without increasing emissions. Compared to the previous scenario, the mixed scenario falls short in reducing tax burden inequality but exhibits promise in limiting greenhouse gas emissions.