The introduction and establishment of nonindigenous species (NIS) through global ship movements poses a significant threat to marine ecosystems and economies. While ballast-vectored invasions have been partly addressed by some national policies and an international agreement regulating the concentrations of organisms in ballast water, biofouling-vectored invasions remain largely unaddressed. Development of additional efficient and cost-effective ship-borne NIS policies requires an accurate estimation of NIS spread risk from both ballast water and biofouling. We demonstrate that the first-order Markovian assumption limits accurate modeling of NIS spread risks through the global shipping network. In contrast, we show that higher-order patterns provide more accurate NIS spread risk estimates by revealing indirect pathways of NIS transfer using Species Flow Higher-Order Networks (SF-HON). Using the largest available datasets of non-indigenous species for Europe and the United States, we then compare SF-HON model predictions against those from networks that consider only first-order connections and those that consider all possible indirect connections without consideration of their significance. We show that not only SF-HONs yield more accurate NIS spread risk predictions, but there are important differences in NIS spread via the ballast and biofouling vectors. Our work provides information that policymakers can use to develop more efficient and targeted prevention strategies for ship-borne NIS spread management, especially as management of biofouling is of increasing concern.