It is found that a cavity design strategy proposed recently to enhance the continuous frequency tuning band of a sub-terahertz second-harmonic gyrotron aggravates the problem of competition between the operating mode and the first-cyclotron-harmonic (fundamental) modes. A stepped coaxial inner conductor is proposed in this article as a means for a selective suppression of the most dangerous fundamental competitors. It is shown that, with a proper choice of the operating mode, the effect of the inner conductor is only distinct for unwanted competing modes. In this case, the gyrotron performance is slightly sensitive to the longitudinal profile of the coaxial insert. Therefore, this profile can be selected in such a way as to reduce the effective cavity length for the competing modes, together with their diffractive quality factors. As a consequence, starting currents of these modes can be selectively increased. This is illustrated by a realistic design of a 0.4-THz second-harmonic gyrotron with 3-GHz continuous frequency tuning.