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The ability to efficiently produce and manipulate nonclassical states of light is a critical requirement for the development of quantum optical technologies. In recent years, experiments have demonstrated that cascaded spontaneous parametric down-conversion is a promising approach to implement photon precertification, providing a way to overcome photon transmission losses for quantum communication, as well as to directly produce entangled three-photon states and heralded Bell pairs. However, the low efficiency of this process has so far limited its applicability beyond basic experiments. Here, we propose a scheme to amplify triplet production rates by using a fast switch and a delay loop to reuse photons that fail to convert on the first pass through the cascade's second nonlinear crystal. We construct a theoretical model to predict amplification rates and verify them in an experimental implementation. Our proof-of-concept device increases the rate of detected photon triplets as predicted, demonstrating that the method has the potential to dramatically improve the usefulness of cascaded down-conversion for device-independent quantum communication and entangled state generation.