Tin-iodide perovskites are an important group of semiconductors for photovoltaic applications, promising higher intrinsic charge-carrier mobilities and lower toxicity than their lead-based counterparts. Controllable tin vacancy formation and the ensuing hole doping provide interesting opportunities to investigate dynamic intraband transitions of charge carriers in these materials. Here, we present for the first time an experimental implementation of a novel Optical-Pump–IR-Push–THz-Probe spectroscopic technique and demonstrate its suitability to investigate the intraband relaxation dynamics of charge carriers brought into nonequilibrium by an infrared “push” pulse. We observe a push-induced decrease of terahertz conductivity for both chemically- and photodoped FA0.83Cs0.17SnI3 thin films and show that these effects derive from stimulated THz emission. We use this technique to reveal that newly photogenerated charge carriers relax within the bands of FA0.83Cs0.17SnI3 on a subpicosecond time scale when a large, already fully thermalized (cold) population of charge-carriers is present. Such rapid dissipation of the initial charge-carrier energy suggests that the propensity of tin halide perovskites toward unintentional self-doping resulting from tin vacancy formation makes these materials less suited to implementation in hot-carrier solar cells than their lead-based counterparts.