5d metals are used in electronics because of their high spin-orbit coupling (SOC) leading to efficient spin-electric conversion. When C60 is grown on a metal, the electronic structure is altered due to hybridisation and charge transfer. In this work we measure the spin Hall magnetoresistance for Pt/C60 and Ta/C60, finding they are up to a factor 6 higher than for the pristine metals, indicating an increase in spin Hall angle of 20-60%. At low fields of 1-30 mT the presence of the C60 increased the anisotropic magnetoresistance by up to 700%. Our measurements are supported by non-collinear Density Functional Theory calculations, which predict a significant SOC enhancement by C60 that penetrates through the Pt layer, concomitant with trends in the magnetic moment of transport electrons acquired via SOC and symmetry breaking. The charge transfer and hybridisation between the metal and the C60 can be controlled by gating, so our results indicate the possibility of dynamically modifying the SOC of thin metals using molecular layers. This could be exploited in spin transfer torque memories and pure spin current circuits.