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We report the effect of high pressure on the superconducting, vortex pinning, and structural properties of a polycrystalline non-centrosymmetric superconductor Re6Hf. The superconducting transition temperature, Tc, reveals a modest decrease as pressure P increases with a slope -0.046 K/GPa (-0.065 K/GPa) estimated from resistivity measurements up to 8 GPa (magnetization measurement ~ 1.1 GPa). Structural analysis up to ~18 GPa reveals monotonic decreases of lattice constant without undergoing any structural transition and a high value of bulk modulus B0= 333.63 GPa, indicating the stability of the structure. Furthermore, the upper critical field and lower critical field at absolute temperature (Hc2(0) & Hc1(0)) decreases slightly from the ambient pressure value as pressure increases up to 2.5 GPa. In addition, up to P ~ 2.5 GPa using thermally activated flux flow of vortices revealed a double linearity field dependence of activation energy of vortices, confirming the coexistence of single and collective pinning vortex states. Moreover, analysis of critical current density using the collective pinning theory showed the transformation of δTc to δl pinning as pressure increases, possibly due to migration of grain boundaries. Besides, the band structure calculations using density functional theory show that density of states decreases modestly with pressure, which may be a possible reason for such a small decrease in Tc by pressure.