Recent comparisons between the stellar haloes of simulated Milky Way-mass galaxies and observations of similar objects have identified a significant tension between the masses of simulated and observed stellar haloes. Simulated stellar haloes appear to have both total masses and surface density profiles 1-2 dex higher than observed galaxies. In this paper, we compare two suites of 15 simulated Milky Way-like galaxies, each drawn from the same initial conditions and simulated with the same hydrodynamical code, but with two different models for feedback from supernovae. We find that the MUGS simulations, which use an older "delayed-cooling" model for feedback, suffer from the same problems as other simulations examined in the literature, with median surface densities well above observational constraints. The MUGS2 simulations, which instead use a new, physically-motivated superbubble model for stellar feedback, have significantly reduced stellar halo masses and surface densities (~25 times lower on average compared to MUGS stellar haloes), and generally match both the median surface density as well as the diversity of structure seen in observed stellar haloes. We examine how feedback produces differences in the assembly of the stellar halo, both through in-situ stars scattered out of the disc by high-redshift mergers and in ex-situ stars stripped from accreted haloes. We conclude that there is no "missing outskirts" problem in cosmological simulations, so long as supernova feedback is modelled in a way that allows it to efficiently regulate star formation in the progenitor environments of stellar haloes.