In this work we investigate experimentally and theoretically the feasibility to enhance the nonradiative lifetime of the upper heavy hole state in Si/SiGe quantum cascade (QC) structures within a diagonal transition design. The problem of fast nonradiative optical phonon scattering makes it impossible to achieve population inversion with recently demonstrated Si/SiGe QC mid-infrared emitter structures, which use vertical transitions between the first two heavy hole states in a single quantum well. We will show that it is possible to prolong the upper state lifetime by more than an order of magnitude in a diagonal transition active region design, that uses transitions between heavy hole ground states of spatially separated quantum wells. Our experimental findings derived from electroluminescence measurements are in good agreement with calculated values based on a 6-band k·p model.