Terahertz applications urgently require high performance and room temperature terahertz sources. The gyrotron based on the principle of electron cyclotron maser is able to generate watt-to-megawatt level terahertz radiation, and becomes an exceptional role in the frontiers of energy, security and biomedicine. However, in normal conditions, a terahertz gyrotron could generate terahertz radiation with high efficiency on a single frequency or with low efficiency in a relatively narrow tuning band. Here a frequency tuning scheme for the terahertz gyrotron utilizing sequentially switching among several whispering-gallery modes is proposed to reach high performance with broadband, coherence and high power simultaneously. Such mode-switching gyrotron has the potential of generating broadband radiation with 100-GHz-level bandwidth. Even wider bandwidth is limited by the frequency-dependent effective electrical length of the cavity. Preliminary investigation applies a pre-bunched circuit to the single-mode wide-band tuning. Then, more broadband sweeping is produced by mode switching in great-range magnetic tuning. The effect of mode competition, as well as critical engineering techniques on frequency tuning is discussed to confirm the feasibility for the case close to reality. This multi-mode-switching scheme could make gyrotron a promising device towards bridging the so-called terahertz gap.