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Energy poverty is a significant social, economic, and health issue which increasingly affects millions of households worldwide. Both climate change and the socio-economic crisis have aggravated this phenomenon, making families unable to keep adequate comfort conditions at home because of economic constraints and/or dwelling inefficiencies. Considering the recent inflation trends, as well as the global effort to reduce the building sector's carbon emissions, energy retrofitting of buildings emerges as the most forward-looking strategy to cope with energy poverty risk. In the case of large building stocks, which are typical for social housing complexes across the EU, deep and fast energy retrofitting might prove challenging, especially considering the resource shortages and disruptions to occupants that may arise. Therefore, this article investigates the relationship between the envelope's insulation ratio and the risk of energy poverty for households. To this end, diverse scenarios are defined, corresponding to progressive increases in the percentage of building envelope that is insulated. The resulting energy needs are calculated for each of them and correlated with local average incomes and relative energy expenses of households. This is tested on an Italian social housing demo case. The results confirm a predictable but not linear correlation between thermal insulation and reduced energy needs for heating, and an interesting side effect on cooling needs for scenarios that perform better in winter. As for income, energy cost has a greater effect on the energy poverty risk when monthly rent is lower, while energy prices have a major role when rent per month is higher. [ABSTRACT FROM AUTHOR]