The nitrogen hyperfine splitting constant of ditertbutyl nitroxide radicals was measured with electron paramagnetic resonance spectroscopy at near-infinite dilution in near-critical and supercritical ethane, as well as in liquid propane, liquid isobutane, and several nonhydrogen bonding liquid solvents. While the measurements in the liquids are described well by the theory of McRae, large deviations from the liquid behavior are observed in supercritical ethane. The deviations are used as a measure of the effective local density of the solvent around the solute. At the two temperatures investigated, T(r) = 1.009 and T(r) = 1.084, the local density enhancement, defined as the ratio of local to bulk densities, exhibits a maximum of about 3 occurring around 112 the critical density. The maximum is removed well from the critical density, where the maximum of the isothermal compressibility is observed. Local density enhancements are short-range effects and do not correlate well with the development of long-range critical phenomena. Local density enhancement data in ethane are compared with the prewetting transition that has been observed in near-critical ultrapure argon.