Measurements of the thermal conductivity of supercritical n-hexane performed in a coaxial cylinder cell operating in steady state conditions are reported. The present 1321 data of the thermal conductivity of n-hexane were carried out along ten quasi-isotherms above the critical temperature. These data cover the temperature range from 508.17 K to 553.00 K and the pressure range 0.l to 10 MPa. An analysis of the various sources of error leads to an estimated uncertainty that do not exceed 4% (95 level of confidence). The parameters of a background equation previously determined from 577 data of the thermal conductivity of gas and liquid n-hexane are reused here in order to analyze the critical enhancement of the thermal conductivity as a function of temperature and density. A set of theoretical Ising-like equations are re-formulated to describe the thermal conductivity critical enhancement along the near-critical isochoric lines in terms of the effective power laws fitting the singular behavior of the needed fluid properties as functions of the finite temperature distance to the critical point. Assuming the knowledge of the regular temperature behavior of the viscosity, the critical enhancement of the thermal conductivity appears then only characterized by a single effective amplitude. Such a separated analysis of the well-defined temperature effects provides the empirical functional form of the density effects, as a 6th-order polynomial density function, which can be normed whatever the temperature. Finally, our complete formulation of the total thermal conductivity describes, within three standard deviations (with SD = 1.0388%), the 1898 thermal conductivity measurements of n-hexane from 293 to 612 K and densities up to 670 kg m−3.Read less <