Transparency associated with a single second mode in a frequency band-gap, caused by orthogonality-breaking-induced multimode interactions, is investigated in a TM-polarized terahertz waveguide with an arbitrary wall profile. It is found that a pure single mode can penetrate through the Bragg gap originating from the lower-mode resonances. We propose a method of achieving bandwidth-agile transparency in a periodic cylindrical waveguide by means of a suitable wall profile. Perturbation theory with respect to the wall corrugation amplitude reveals the underlying physics of the single-mode transparency, in that the major Fourier component of the wall profile affects the width of the stopband proportionally, whereas the transparent passband within the stopband varies in inverse proportion. Numerical simulations of four different types of wall profile confirm that the width of the transparent band can be manipulated as proposed. When other Fourier components are present, simulations demonstrate only a small variation of the frequency band structure around 1 THz, indicating the validity of the proposed method. More extensive investigations of single-mode transparency will provide significant benefits in future mode-control engineering of terahertz devices.