Explaining why some species are widespread, while others are not, is fundamental to biogeography, ecology, and evolutionary biology. A unique way to study evolutionary and ecological mechanisms that either limit species' spread or facilitate range expansions is to conduct research on species that have restricted distributions. Nonindigenous species, particularly those that are highly invasive but have not yet spread beyond the introduced site, represent ideal systems to study range size changes. Here, we used species distribution modeling and genomic data to study the restricted range of a highly invasive Australian marine species, the ascidian Pyura praeputialis. This species is an aggressive space occupier in its introduced range (Chile), where it has fundamentally altered the coastal community. We found high genomic diversity in Chile, indicating high adaptive potential. In addition, genomic data clearly showed that a single region from Australia was the only donor of genotypes to the introduced range. We identified over 3,500 km of suitable habitat adjacent to its current introduced range that has so far not been occupied, and importantly species distribution models were only accurate when genomic data were considered. Our results suggest that a slight change in currents, or a change in shipping routes, may lead to an expansion of the species' introduced range that will encompass a vast portion of the South American coast. Our study shows how the use of population genomics and species distribution modeling in combination can unravel mechanisms shaping range sizes and forecast future range shifts of invasive species.