Besides the two fundamental symmetric and antisymmetric modes, a thin metal slab surrounded by a dielectric medium supports an infinite number of other modes whose propagation distances, however, are extremely short. We show that such a higher-order mode, normally not regarded to be useful, can turn into a long-range surfaceplasmon polariton in circumstances where the fundamental long-range mode does not exist and the propagation length of the single-interface surface-plasmon polariton is negligible. This transition, which is an electromagnetic near-field phenomenon not previously observed, occurs with a broad range of bandwidths and materials, especially in the context of high-index semiconductors. The long-range behavior of the higher-order surface-plasmon polaritons does not originate from the coupling between the polaritons supported by the individual boundaries, and in some situations it comes with increased field confinement, which is the opposite of what occurs for the fundamental long-range surface mode. Our results, which follow from rigorous electromagnetic theory, provide deeper insight into the foundations of thin-film plasmonics and suggest avenues for enhanced properties of nanophotonic devices involving modern polaritonic media.