We derive a family of local-density-specific radial distribution functions which enable us to analyze the fluid structure around a tagged solvent particle as a function of the local environment about that particle. With this new structural measure, we uncover a very strong correlation between the local density around a tagged atom and the surrounding long-range fluid structure in a neat, two-dimensional, Lennard-Jones fluid at a compressible supercritical state point. Indeed, we show that such an environmentally specific measure of structure provides an informative probe of compressible supercritical fluids, within which the distribution of accessible local solvent environments is typically very broad. Additionally, we find that these new functions provide a more sensitive measure of the mean size of the density inhomogeneities than does the correlation length, and thus that they provide an effective indicator of simulation convergence for molecular dynamics simulations performed in the supercritical regime.