Stoichiometric SnTe is theoretically a small gap semiconductor that undergoes a ferroelectric distortion on cooling. In reality however, crystals are always nonstoichiometric and metallic; the ferroelectric transition is therefore, more accurately described as a polar structural transition. Here, we study the Fermi surface using quantum oscillations as a function of pressure. We find the oscillation spectrum changes at high pressure due to the suppression of the polar transition and less than 10 kbars is sufficient to stabilize the undistorted cubic lattice, this is accompanied by a large decrease in the Hall and electrical resistivities. Combined with our density functional theory calculations and angle-resolved photoemission spectroscopy measurements, this suggests the Fermi surface L pockets have lower mobility than the tubular Fermi surfaces that connect them. Additionally, we find the unusual phenomenon of a linear magnetoresistance that exists irrespective of the distortion that we attribute to regions of the Fermi surface with high curvature.