Numerous studies have shown that a prior SARS-CoV-2 infection can greatly enhance the antibody response to COVID-19 vaccination, with this so called "hybrid immunity" leading to greater neutralization breadth against SARS-CoV-2 variants of concern. However, little is known about how breakthrough infection (BTI) in COVID-19-vaccinated individuals will impact the magnitude and breadth of the neutralizing antibody response. Here, we compared neutralizing antibody responses between unvaccinated and COVID-19-double-vaccinated individuals (including both AZD1222 and BNT162b2 vaccinees) who have been infected with the Delta (B.1.617.2) variant. Rapid production of spike-reactive IgG was observed in the vaccinated group, providing evidence of effective vaccine priming. Overall, potent cross-neutralizing activity against current SARS-CoV-2 variants of concern was observed in the BTI group compared to the infection group, including neutralization of the Omicron (B.1.1.529) variant. This study provides important insights into population immunity where transmission levels remain high and in the context of new or emerging variants of concern.