We study the local solute-solvent structure in dilute supercritical solutions, using as a model system a dilute Yukawa solute in a supercritical Lennard-Jones fluid. Our primary interest is in the effect of the solute-solvent interaction range on the local solvent density around the solute. We employ the integral equation theory for inhomogeneous fluids to calculate the solute-solvent structural properties. The theory is shown to be in excellent agreement with Monte Carlo simulations and to provide a substantial improvement over the integral equation theory formulated for homogeneous fluids. In particular, it is demonstrated that the homogeneous theory greatly overestimates the local density enhancement for long-ranged solute-solvent interactions in the highly compressible supercritical regime.