A compact coaxial relativistic klystron amplifier (RKA) with three cascaded bunching cavities is investigated to obtain high efficiency output in this paper. When the injection power and drift tube length are both decreased for the compactness of the initial coaxial RKA, the conversion efficiency of this device decreases to only 20%, which is mainly due to the insufficient modulation of intense relativistic electron beam (IREB). To solve this issue, three cascaded bunching cavities are designed to strengthen the modulation of IREB, and each bunching cavity is designed to a non-uniform single-gap cavity with a high external quality factor. Moreover, a specific drift tube length is chosen for minimizing the TEM mode leakage between every two adjacent cavities. With these methods, the self-oscillation between input and three cascaded bunching cavities can be successfully suppressed without loading any reflectors. The proposed three cascaded single-gap bunching cavities are further examined in a Ku-band coaxial RKA, and the output efficiency of this device increases to 44%, which is more than twice that of the initial coaxial RKA. Furthermore, the injection power and axial length of the propagating IREB in the improved coaxial RKA are reduced to only 5 kW and 10 wavelengths, respectively.