We make use of the IllustrisTNG cosmological, hydrodynamical simulations to
test fundamental assumptions of the mass-based Halo Occupation Distribution
(HOD) approach to modelling the galaxy-halo connection. By comparing the
clustering of galaxies measured in the 300 Mpc TNG box (TNG300) with that
predicted by the standard ("basic") HOD model (bHOD), we find that, on average,
the bHOD model underpredicts the real-space correlation function in the TNG300
box by $\sim$ 15\% on scales of $1 \ {\rm Mpc}/h < r < 20 \ {\rm Mpc}/h$, which
is well beyond the target precision demanded of next-generation galaxy redshift
surveys. We perform several tests to establish the robustness of our findings
to systematic effects, including the effect of finite box size and the choice
of halo finder. In our exploration of "secondary" parameters with which to
augment the bHOD, we find that the local environment of the halo, the velocity
dispersion anisotropy, $\beta$, and the product of the half-mass radius and the
velocity dispersion, $\sigma^2 R_{\rm halfmass}$, are the three most effective
measures of assembly bias that help reconcile the bHOD-predicted clustering
with that in TNG300. In addition, we test other halo properties such as halo
spin, formation epoch and halo concentration. We also find that at fixed halo
mass, galaxies in one type of environment cluster differently from galaxies in
another. We demonstrate that a more complete model of the galaxy-halo
connection can be constructed if both the mass and information regarding the
local environment in which a halo is embedded are combined.