Most studies on seasonal influenza A/H3N2 virus adaptation have focused on the main antigenic gene, hemagglutinin. However, there is increasing evidence that the genome-wide genetic background of novel antigenic variants can influence these variants' emergence probabilities and impact their patterns of dominance in the population. This suggests that non-antigenic genes may be important in shaping the viral evolutionary dynamics. To better understand the role of selection on non-antigenic genes in the adaptive evolution of seasonal influenza viruses, we have developed a simple population genetic model that considers a virus with one antigenic and one non-antigenic gene segment. By simulating this model under different regimes of selection and reassortment, we find that the empirical patterns of lineage turnover for the antigenic and non-antigenic gene segments are best captured when there is both limited viral coinfection and selection operating on both gene segments. In contrast, under a scenario of only neutral evolution in the non-antigenic gene segment, we see persistence of multiple lineages for long periods of time in that segment, which is not compatible with observed molecular evolutionary patterns. Further, we find that reassortment, occurring in coinfected individuals, can increase the speed of viral adaptive evolution by primarily reducing selective interference and genetic linkage effects. Together, these findings suggest that, for influenza, with six internal or non-antigenic gene segments, the evolutionary dynamics of novel antigenic variants are likely to be influenced by the genome-wide genetic background as a result of linked selection among both beneficial and deleterious mutations.