In this study, the first- (K1) and second-order (K2) magnetic anisotropies are investigated as a function of post-annealing temperature (Ta) in Pt/Co/MgO heterostructures. We find that both extrinsic and intrinsic mechanisms contribute to K2 with their relative contributions significantly depending on the quality of the Co/MgO interface, which is sensitively affected by the Ta value. In contrast with previous studies that mainly considered the extrinsic effects on K2, we obtain high K2 values of up to 2.04 × 106 erg/cm3 for the stack annealed at Ta = 400 °C by maximally utilizing the intrinsic effect at the Co/MgO interface, thus facilitating robust easy-cone anisotropy. We also demonstrate that the canted magnetization can be efficiently manipulated by the spin–orbit torques generated from the Pt layer, which is an important step toward the application of easy-cone states in various spintronic devices.