Abstract Background Increasing vaccination coverage against SARS-CoV-2 enabled relaxation of lockdowns in many countries in Europe. As the vaccination rollouts progressed, the public health authorities were seeking recommendations on the continuation of physical distancing measures during ongoing vaccination rollouts. Compliance with these measures was declining while more transmissible virus variants have emerged. Methods We used a SARS-CoV-2 transmission model to investigate the feedback between compliance, infection incidence, and vaccination coverage. We quantified our findings in terms of cumulative number of new hospitalisations three and six months after the start of vaccination. Results Our results suggest that the combination of fast waning compliance in non-vaccinated individuals, low compliance in vaccinated individuals, low vaccine efficacy against infection and more transmissible virus variants may result in a higher cumulative number of new hospitalisations than in a situation without vaccination. These adverse effects can be alleviated by deploying behavioural interventions that should preferably target both vaccinated and non-vaccinated individuals. The choice of the most appropriate intervention depends on vaccination rate and vaccine efficacy against infection. Conclusions Supplementary behavioural interventions aiming to boost compliance to physical distancing measures can improve the outcome of vaccination programmes, until vaccination coverage is sufficiently high. For optimal results, these interventions should be selected based on the vaccine efficacy against infection and expected vaccination rate. While we considered the dynamics of SARS-CoV-2, the qualitative effects of the interplay between infectious disease spread and behavior on the outcomes of a vaccination programme can be used as guidance in a future similar pandemic.