Quantum mechanical confinement in electron-hole bilayer tunnel field-effect transistors (EHBTFETs) affects in a substantial way the band-to-band tunneling (BTBT) mechanism that constitutes their operating principle. Field-induced quantization is known to set off effective bandgap widening phenomena-with the subsequent BTBT probability reduction that it entails-and to give rise to harmful parasitic tunneling processes. Both of these effects degrade the potential steep switching behavior of bilayer TFETs. In this paper, we show that the novel FinEHBTFET proves to be a promising structure for its scalability potential and propose a solution to alleviate the impact on it of quantum confinement, as well as to suppress the parasitic tunneling processes that show up when quantization is considered. Moreover, we demonstrate for different fin materials that the utilization of asymmetric configurations delaying the formation of electron inversion layers allows us to boost ION levels.