After ITER, the first real prototype of a fusion power plant is foreseen. It is named Demonstration Power Plant (DEMO). Within the EUROfusion work package "Heating and Current Drive", the conceptual designs of both a hollow-cavity and coaxial-cavity gyrotron for DEMO has been suggested. For DEMO, the major target is to achieve an output power per tube which is significantly larger than 1 MW CW at an operating frequency of up to 240 GHz. In the case of the hollow-cavity gyrotron design, a very high-order TE mode (eigenvalue > 103 at a cutoff frequency of 236 GHz) is the prerequisite for the targeted high output power. At this eigenvalue range, the excitation of the desired operating mode during gyrotron start-up becomes challenging due to a large number of competing modes. In this work, it is demonstrated by numerical simulation that mode excitation and mode competition control can be significantly facilitated by using special start-up scenarios, based on a triode electron gun.