Topological crystalline insulator tin telluride (SnTe) provides a rich playground to examine interactions of correlated electronic states, such as ferroelectricity, topological surface states, and superconductivity. The study of SnTe nanowires may lead to even richer physics owing to the one-dimensional (1D) confinement effect and an increased contribution from the topological surface states. Thus, for transport measurements, SnTe nanowires must be synthesized with reduced diameters and high crystalline quality to ensure 1D confinement and phase coherence of the topological surface electrons. This study reports a facile growth method to produce narrow SnTe nanowires with a high yield using alloy nanoparticles as growth catalysts. The average diameter of the SnTe nanowires grown using alloy nanoparticles is 85 nm, nearly a factor of three reduction compared to the average diameter of 240 nm when using gold nanoparticles as growth catalysts. Transport measurements reveal the effect of the nanowire diameter on the residual resistance ratio and magnetoresistance. Particularly, the ferroelectric transition temperature for SnTe evolves systematically with the nanowire diameter. In situ cryogenic cooling of narrow SnTe nanowires in a transmission electron microscope directly reveals the cubic to rhombohedral structural transition, which is associated with the ferroelectric transition. Thus, these narrow SnTe nanowires represent a model system to study electronic states arising from 1D confinement, such as 1D topological superconductivity and potential multiband superconductivity.