Despite the increasing biological and economic impacts of invasive species, little is known about the evolutionary mechanisms that favor geographic range expansion and evolution of invasiveness in introduced species. Here, we focus on the invasive wetland grass Phalaris arundinacea L. and document the evolutionary consequences that resulted from multiple and uncontrolled introductions into North America of genetic material native to different European regions. Continental-scale genetic variation occurring in reed canarygrass' European range has been reshuffled and recombined within North American introduced populations, giving rise to a number of novel genotypes. This process alleviated genetic bottlenecks throughout reed canarygrass' introduced range, including in peripheral populations, where depletion of genetic diversity is expected and is observed in the native range. Moreover, reed canarygrass had higher genetic diversity and heritable phenotypic variation in its invasive range relative to its native range. The resulting high evolutionary potential of invasive populations allowed for rapid selection of genotypes with higher vegetative colonization ability and phenotypic plasticity. Our results show that repeated introductions of a single species may inadvertently create harmful invaders with high adaptive potential. Such invasive species may be able to evolve in response to changing climate, allowing them to have increasing impact on native communities and ecosystems in the future. More generally, multiple immigration events may thus trigger future adaptation and geographic spread of a species population by preventing genetic bottlenecks and generating genetic novelties through recombination.