Hybrid semiconductor-superconductor nanowires constitute a pervasive platform for studying gate-tunable superconductivity and the emergence of topological behavior. Their low dimensionality and crystal structure flexibility facilitate unique heterostructure growth and efficient material optimization, crucial prerequisites for accurately constructing complex multicomponent quantum materials. Here, we present an extensive study of Sn growth on InSb, InAsSb, and InAs nanowires and demonstrate how the crystal structure of the nanowires drives the formation of either semimetallic α-Sn or superconducting β-Sn. For InAs nanowires, we observe phase-pure superconducting β-Sn shells. However, for InSb and InAsSb nanowires, an initial epitaxial α-Sn phase evolves into a polycrystalline shell of coexisting α and β phases, where the β/α volume ratio increases with Sn shell thickness. Whether these nanowires exhibit superconductivity or not critically relies on the β-Sn content. Therefore, this work provides key insights into Sn phases on a variety of semiconductors with consequences for the yield of superconducting hybrids suitable for generating topological systems.