Context. Massive star evolution plays a crucial role in astrophysics; however, its study is subject to large uncertainties. This problem becomes more severe by the majority of massive stars being born in close binary systems, whose evolution is affected by interactions among their components. Aims. We want to constrain major uncertainties in massive binary star evolution, particularly with respect to the efficiency and the stability of the first mass-transfer phase. Methods. We used the rapid population synthesis code COMBINE to generate synthetic populations of post-interaction binaries, assuming constant mass-transfer efficiency. We employed a new merger criterion that adjusts self-consistently to any prescribed mass-transfer efficiency. We tailored our synthetic populations to be comparable to the expected binary populations in the Small Magellanic Cloud (SMC). Results. We find that the observed populations of evolved massive binaries cannot be reproduced with a single mass-transfer efficiency. Instead, a rather high efficiency (≳50%) is needed to reproduce the number of Be stars and Be/X-ray (BeXB) binaries in the SMC, while a low efficiency (∼10%) leads to a better agreement with the observed number of Wolf-Rayet (WR) stars. We constructed a corresponding mass-dependent mass-transfer efficiency recipe to produce our fiducial synthetic SMC post-interaction binary population. It reproduces the observed number and properties of the BeXBs and WR binaries rather well; furthermore, it is not in stark disagreement with the observed OBe star population. It predicts around 170 massive stars with neutron star companion, of which 140 are Be stars, and about 170 systems disrupted by the supernova, of which 150 are Be stars. Overall, 20% of all post-interaction systems contain a helium star. It also predicts two large, as-yet-unobserved populations of OB + BH binaries: about 100 OB + BH systems with rather small orbital periods (≲20 d) and around 40 longer period OBe + BH systems. Conclusions. Continued searches for massive binary systems will strongly advance our understanding of their evolution.