We study the thermocapillary-driven spreading of a droplet on a nonuniformly heated substrate for fluids associated with a non-monotonic dependence of the surface tension on temperature. We use lubrication theory to derive an evolution equation for the interface that accounts for capillarity and thermocapillarity. The contact line singularity is relieved by using a slip model and a Cox-Voinov relation; the latter features equilibrium contact angles that vary depending on the substrate wettability, which, in turn, is linked to the local temperature. We simulate the spreading of droplets of fluids whose surface tension temperature curves exhibit a turning point. For cases wherein these turning points correspond to minima, and when these minima are located within the droplet, then thermocapillary stresses drive rapid spreading away from the minima. This gives rise to a significant acceleration of the spreading whose characteristics resemble those associated with the "superspreading" of droplets on hydrophobic substrates. No such behavior is observed for cases in which the turning point corresponds to a surface tension maximum.