Tin telluride (SnTe) has long been known to undergo a pressure-induced semiconductor–superconductor transition, but an understanding of the underlying mechanism has been impeded by unsettled issues concerning its structural identification and phase boundary at high pressure. A recent X-ray diffraction measurement combined with a first-principles structural search has resolved these structural issues, making it possible to explore further this intriguing material. Here, we report on a first-principles study of the electronic band structure and superconductivity of SnTe in a wide pressure range. We obtain a superconducting state in the high-pressure B2 (Pm-3m) phase of SnTe with a maximum critical temperature of 7.16 K, which is in excellent agreement with the experimental value of 7.5 K. Moreover, we find additional superconducting states in two previously unidentified intermediate phases of orthorhombic Cmcm and Pnma symmetry. Our analysis indicates that superconductivity in the intermediate phases of SnTe is closely correlated with a pressure-driven evolution of the topological nature of the Fermi surface that has a large impact on the electron–phonon coupling. Meanwhile, the much higher TC of the high-pressure B2 phase stems from the simultaneous presence of highly mobile and extremely confined conduction electrons, which enhances electron pairing and superconductivity. These results provide insights into pressure-induced superconductivity and electronic phase transitions in SnTe and have implications for understanding other narrow-bandgap IV–VI semiconductors at high pressure.