Considering 2→2 gauge-theory scattering with general colour in the high-energy limit, we compute the Regge-cut contribution to three loops through next-to-next-to-leading high-energy logarithms (NNLL) in the signature-odd sector. Our formalism is based on using the non-linear Balitsky-JIMWLK rapidity evolution equation to derive an effective Hamiltonian acting on states with a fixed number of Reggeized gluons. A new effect occurring first at NNLL is mixing between states with k and k+2 Reggeized gluons due non-diagonal terms in this Hamiltonian. Our results are consistent with a recent determination of the infrared structure of scattering amplitudes at three loops, as well as a computation of 2→2 gluon scattering in N=4 super Yang-Mills theory. Combining the latter with our Regge-cut calculation we extract the three-loop Regge trajectory in this theory. Our results open the way to predict high-energy logarithms through NNLL at higher-loop orders.