We use the improved IllustrisTNG300 magneto-hydrodynamical cosmological simulation to revisit the effect that secondary halo bias has on the clustering of the central galaxy population. With a side length of 205 $h^{-1}$Mpc and significant improvements on the sub-grid model with respect to the previous Illustris boxes, IllustrisTNG300 allows us to explore the dependencies of galaxy clustering over a large cosmological volume and wide halo-mass range. We show, at high statistical significance, that the halo assembly bias signal (i.e., the secondary dependence of halo bias on halo formation redshift) manifests itself on the clustering of the central galaxy population when this is split by stellar mass, colour, specific star formation rate, and surface density. A significant detection is also obtained for galaxy size: at fixed halo mass, larger central galaxies are more tightly clustered than smaller central galaxies in haloes of mass M$_{rm vir} lesssim 10^{12.5}$ $h^{-1}$M$_{odot}$. This effect, however, seems to be uncorrelated with halo formation time, unlike the rest of the secondary dependencies analysed. We also explore the transmission of the halo spin bias signal, i.e., the secondary dependence of halo bias on halo spin. Although galaxy spin retains little information about the total spin of the halo, the correlation is enough to produce a significant galaxy spin bias signal. We discuss possible ways to probe the spin bias effects with observations.
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