Reducing afterpulsing in single-photon avalanche diodes (SPADs) allows operation with shorter recovery times and higher detection rates. Afterpulsing in SPADs can be reduced by reducing the total avalanche charge. We use a periodic quenching system to arbitrarily vary the latency between the onset of an avalanche and the application of the quench, allowing us to characterize the afterpulsing behavior when the current flow is halted at time scales that are significantly shorter than can be achieved by standard active-quenching systems. Three different reach-through SPADs are characterized, and with prompt quenching we observe reductions in afterpulse probability of as much as a factor of 12. Beyond improving detection rates, reducing the total avalanche charge can also allow operation with higher excess bias voltages, which enables higher detection efficiency and more precise timing resolution.