We measured the temperature dependent resistivity, varrho(T), of the intercalated graphite superconductor CaC6 as a function of pressure up to 16 GPa. We found a large linear increase of critical temperature, Tc, from the ambient pressure value 11.5 K up to 15.1 K, the largest value for intercalated graphite, at 7.5 GPa. At approximately 8 GPa, a jump of varrho and a sudden drop of Tc down to approximately 5 K indicates the occurrence of a phase transition. Our data analysis suggests that a pressure-induced phonon softening related to an in-plane Ca phonon mode is responsible for the Tc increase and that higher pressures greater, similar8 GPa lead to a structural transition into a new phase with a low Tc less, similar3 K.