•High-pressure and high-temperature PVT relationship of the dilute n-hexane+ methyl octanoate mixture.•Krichevskii parameter of the n-hexane+ methyl octanoate mixture.•The critical parameters of the n-hexane+ methyl octanoate mixture.•Bubble and dew-point curves of the n-hexane+ methyl octanoate mixture. PρT and phase transition properties (PS,ρS,TS)  of the dilute binary mixture of n-hexane + methyl octanoate have been studied near the critical point of pure n-hexane (solvent). The measurements were performed along 11 liquid, vapor and near-critical isochores between (39.3 and 621.1) kg·m-3 over a temperature range from (293 to 622) K at pressures up to 17.7 MPa for the fixed concentration of x = 0.088 mole fraction (or 0.05 mass fraction) of methyl octanoate. The measurements were made using a constant-volume piezometer technique. The study was concentrated in the single - and two- phase regions near the critical point of pure n-hexane to precisely determine the phase boundary properties (PS,ρS,TS) and the critical property (PC,ρC,TC) data of the mixture. The measured critical property data for the mixture were used to study of the isomorphic critical behavior of strongly singular properties such as isothermal compressibility KTX and isobaric heat capacity CPX along the critical isochore and the critical isotherm, i.e., Fisher renormalization (mixture-like behavior) of the critical exponent, -γ⇒γ/(1−α), for strongly singular properties of mixture under study. The measured PρT data were also used to estimate the values of physical meaningful (theoretically important) parameters such as characteristic reduced temperature τ1and density differencesΔρ1 of the mixture. We found that KTX and CPX of the studied mixture exhibit pure-like behavior along the critical isochore at temperature above 508.52 K (slightly above the critical temperature of pure n-hexane, 507.44 K) and along the critical isotherm at densities above 279.82 kg⋅m-3 (above the critical density of pure n-hexane, 232.8 kg⋅m-3). The measured data were used to estimate the Krichevskii parameter Kkr,structural (Nexc)and volumetric (V¯2∞) properties of the dilute mixture near the critical point of pure solvent (n-hexane).