Pathogen adaptation to public health interventions, such as vaccination, may take tortuous routes and involve a multitude of genetic mutations acting on distinct phenotypic traits. For example, pathogens can escape the vaccine-induced immune response, or adjust their virulence so as to increase transmission in vaccinated hosts. Despite its importance for public health and vaccine efficacy, how these two adaptations jointly evolve is poorly understood. Taking a trait-centered, rather than variant-centered perspective, here we elucidate the role played by epistasis and recombination, with an emphasis on the different protective effects of vaccination. Vaccines reducing transmission and/or increasing clearance generate positive epistasis between the vaccine-escape and virulence alleles, favouring strains that carry both mutations. Vaccines reducing virulence mortality generate negative epistasis, favouring strains that carry either mutation, but not both. If epistasis is positive, recombination can lead to the sequential fixation of the two mutations and prevent the transient build-up of more virulent escape strains. If epistasis is negative, recombination between loci can create an evolutionary bistability between alternative routes of adaptation, such that whichever adaptation is more accessible tends to be favoured in the long-term. More generally, our model provides a valuable framework for studying pathogen adaptation from a trait-centered view, shifting the focus away from variants.