We compile a large sample of nearby galaxies that are satellites of hosts using a well-known Sloan Digital Sky Survey (SDSS) group catalog. From this sample, we create an "ancient infallers" and "recent infallers" subsample, based on the mean infall time predicted from cosmological simulations for galaxies with their location in phase-space. We compare the stellar mass growth histories of the galaxies in these two subsamples, as determined from multiwavelength spectral energy distribution fitting that uses a comprehensive library of star formation history shapes derived from cosmological simulations. By simultaneously controlling for satellite stellar mass and host halo mass, we can clearly see the impact of time spent in their hosts. As we might predict, the ancient infaller population shows clear signs of earlier quenching, especially for lower mass satellites in more massive hosts. More importantly, we find that the effects are not limited to massive hosts. We find that hosts that might be considered low-mass groups (halo masses similar to 10(13) M-circle dot) can significantly alter their satellites, even for massive satellites (stellar masses similar to 10(10) M-circle dot). Intriguingly, we see changes in the mass growth history of the satellites of clusters as early as 8 or 9 Gyr ago, when they had not yet entered the virial radius of their current host. We propose that this could be the result of galaxies being preprocessed in low-mass substructures in the protocluster outskirts, prior to infall.