For most of the applications in the millimeter wave band, corresponding to Ka and higher-frequency bands, the relatively high atmospheric absorption necessitates the use of high-power sources. Here, a new approach for projecting an oversized beam tunnel in an overmoded structure by concentrating the axial field is demonstrated to meet the high-frequency and high-power demands of compact devices. Due to the enhanced high-current intense beam loading capability of the interaction circuit, a six-cavity Ka-band extended interaction klystron (EIK) with a four-coupling-hole disk-loaded structure is designed that can stably obtain high output power. An analysis of optimization trade-offs from introducing high order modes for allowing the application of more powerful beams to improving high order modes field distribution for enhancing the electron-wave coupling and suppressing mode competition is reported. 3-D particle in cell (PIC) simulations show that such new design approaches can achieve an output power of 1.11 MW at 32.94 GHz with a saturated gain of 57 dB by injecting a 3.3 mm diameter electron beam with a current of 24 A.