Energy market is witnessed to have high integration with international equity markets ranging from traditional assets to newly introduced asset classes, especially after increased financialization during the last decade. We compare the connectedness between international oil market and socially responsible stocks before and during the recent pandemic COVID-19. Socially responsible stocks are increasingly becoming popular because of their social cause as well as less integration with other assets, however since their acceptance, have never been tested against the contagion phenomena. We utilize daily data ranging from March 2016 to June 2020 to investigate integration and spillover before and during COVID-19 period. Our methodological framework includes asymmetric multifractal detrended approach, network connectedness and conditional diversification measures. Our sampled SRI funds exhibit multifractal behaviour and share bidirectional spillover with oil. The SRI funds demonstrate good hedging abilities during normal and COVID-19 period and for similar reason are suggested as optimal asset together with oil futures in a portfolio. Furthermore, these SRI funds yield optimal diversification benefits with oil futures under bearish market conditions.