We present results of an experiment showing the first successful demonstration of a cascaded microbunching scheme. Two modulator-chicane prebunchers arranged in series and a high power mid-IR laser seed are used to modulate a 52 MeV electron beam into a train of sharp microbunches phase locked to the external drive laser. This configuration is shown to greatly improve matching of the beam into the small longitudinal phase space acceptance of short-wavelength accelerators. We demonstrate trapping of nearly all (96%) of the electrons in a strongly tapered inverse free-electron laser accelerator, with an order-of-magnitude reduction in injection losses compared to the classical single-buncher scheme. These results represent a critical advance in laser-based longitudinal phase space manipulations and find application in high gradient advanced acceleration as well as in high peak and average power coherent radiation sources.