In previous work, we showed that a semi-empirical model in which galaxies in host dark matter haloes are assigned stellar masses via a stellar mass–halo mass (SMHM) relation and sizes (R$_e$) via a linear and tight R$_e$–R$_h$ relation can faithfully reproduce the size function of local Sloan Digital Sky Survey (SDSS) central galaxies and the strong size evolution of massive galaxies (MGs; M$_{star}$ > 10$^{11.2}$M$_⊙$). In this third paper of the series, we focus on the population of satellite MGs. We find that without any additional calibration and irrespective of the exact SMHM relation, fraction of quenched galaxies, or level of stellar stripping, the same model is able to reproduce the local size function of quiescent satellite MGs in SDSS. In addition, the same model can reproduce the puzzling weak dependence of mean size on host halo mass for both central and satellite galaxies. The model also matches the size function of star-forming satellite MGs, after assuming that some of them transform into massive lenticulars in a few Gyr after infalling in the group/cluster environment. However, the vast majority of satellite lenticulars are predicted to form before infall. The R$_e$–R$_h$ relation appears to be fundamental to connect galaxies and their host haloes.