In this paper, we use the method of high order TMn1 mode selection from the concept of narrow-band Smith-Purcell radiation (SPR) for powerful, over-mode, multi-gap extended interaction circuit designs toward millimeter wave and Terahertz (THz) region. As a core part, the multiple gaps interaction structure, equivalent to a subwavelength hole array (SHA), excites the narrow band SPR when an electron beam is injected. The SPR energy is collected by a pair of closed cavities, which satisfies (n-1) standing wave units. The SPR energy in the optimized cavity allows a high index n TMn1 mode operation to achieve the strongest Ez field and high characteristic impedance in a closed multi-gap resonant circuit. This provides an effective design to establish a stable high-order TMn1 mode that supports extended interaction circuits with large cross sections. A 0.46 THz extended interaction circuit, employing the novel high order TM51-2π mode operation output structure, has been designed to demonstrate the efficient beam-wave interaction in the proposed system. The method of TMn1 mode selection provides new insight into the understanding of the high-frequency extended interaction circuits by introducing the SPR concept, benefiting the development of millimeter wave and THz vacuum electron devices (VEDs).