In order to study parasitic oscillation that may occur in a realistic beam duct upstream to the gyrotron cavity, the self-consistent linear and spectral code TWANGlinspec has been modified. The large inhomogeneities in the smooth-wall beam duct geometry or in the magnetic field profile required the implementation of a numerical approach using a hybrid finite element method. The new model permits to characterize a large number of potentially spurious TE modes. Compared to previous studies on gyrotron beam duct instabilities, an extended interaction space including also the gyrotron cavity has been considered. The role of the connecting part between the beam duct and the cavity, called spacer, is highlighted and it is shown that the gyro backward-wave TE modes excited in this region generally have their minimum starting current. The sensitivity of the minimum starting current on electron beam velocity spread is also evaluated.