We investigate the tunneling anisotropic magnetoresistance (TAMR) in thick hcp Co films at cryogenic temperatures using scanning tunneling microscopy. At around -350 mV, a strong TAMR up to 30 per cent is found with a characteristic voltage dependence and a reversal of sign. With the help of ab initio calculations, the TAMR can be traced back to a spin-polarized occupied surface state that experiences a strong spin-orbit interaction leading to a magnetization direction depending on hybridization with bulk states.