Europe has experienced a large COVID-19 wave caused by the Delta variant in winter 2021/22. Using mathematical models applied to Metropolitan France, we find that boosters administered to ≥ 65, ≥ 50 or ≥ 18 year-olds may reduce the hospitalisation peak by 25%, 36% and 43% respectively, with a delay of 5 months between second and third dose. A 10% reduction in transmission rates might further reduce it by 41%, indicating that even small increases in protective behaviours may be critical to mitigate the wave.