  Considering $2\to 2$ 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 $2\to 2$ gluon scattering in ${\cal N}=4$
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.
