PbTe is a leading mid-range thermoelectric material with a $zT$ that has been enhanced by, amongst other methods, band engineering. Here we present an experimental study of the Hall effect, quantum oscillations, specific heat, and electron microprobe analysis that explores the evolution of the electronic structure of PbTe heavily doped with the `ideal' acceptor Na up to the solubility limit. We identify two phenomenological changes that onset as the electronic structure deviates from a Kane-type dispersion at around 180meV; a qualitative change in the field dependence of the Hall effect indicative of an increase in the high-field limit and a change in the Fermiology, and a sharp increase in the density of states as a function of energy. Following consideration of three possible origins for the observed phenomenology we conclude that the most likely source is non-ellipsoidicity of the $L$-pocket upon approach to a saddle point in the band structure, which is evidenced directly by our quantum oscillation measurements. Comparison to density functional theory calculations imply that this evolution of the electronic structure may be a key contributor to the large thermopower in PbTe.