In this paper we extend the results in [16] to more general domains. More precisely, we obtain transmission eigenvalue-free regions for the interior transmission problem with one complex-valued refraction index, that is, with a damping term which does not vanish on the boundary. In particular, we remove the non-trapping condition as well as the strict concavity condition from [16]. Instead, we impose new, more general conditions in terms of the highfrequency behavior of certain cut-off resolvents associated to exterior problems.