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There is untapped cosmological information in galaxy redshift surveys in the nonlinear regime. In this work, we use the Aemulus suite of cosmological N -body simulations to construct Gaussian process emulators of galaxy clustering statistics at small scales (0.1–50 h ^−1 Mpc) in order to constrain cosmological and galaxy bias parameters. In addition to standard statistics—the projected correlation function w _p ( r _p ), the redshift-space monopole of the correlation function ξ _0 ( s ), and the quadrupole ξ _2 ( s )—we emulate statistics that include information about the local environment, namely the underdensity probability function P _U ( s ) and the density-marked correlation function M ( s ). This extends the model of Aemulus III for redshift-space distortions by including new statistics sensitive to galaxy assembly bias. In recovery tests, we find that the beyond-standard statistics significantly increase the constraining power on cosmological parameters of interest: including P _U ( s ) and M ( s ) improves the precision of our constraints on Ω _m by 27%, σ _8 by 19%, and the growth of structure parameter, f σ _8 , by 12% compared to standard statistics. We additionally find that scales below ∼6 h ^−1 Mpc contain as much information as larger scales. The density-sensitive statistics also contribute to constraining halo occupation distribution parameters and a flexible environment-dependent assembly bias model, which is important for extracting the small-scale cosmological information as well as understanding the galaxy–halo connection. This analysis demonstrates the potential of emulating beyond-standard clustering statistics at small scales to constrain the growth of structure as a test of cosmic acceleration.