  We study a family of higher-twist Regge trajectories in $\mathcal{N}=4$
supersymmetric Yang-Mills theory using the Quantum Spectral Curve. We explore
the many-sheeted Riemann surface connecting the different trajectories and show
the interplay between the degenerate non-local operators known as horizontal
trajectories. We resolve their degeneracy analytically by computing the first
non-trivial order of the Regge intercept at weak coupling, which exhibits new
behaviour: it depends linearly on the coupling. This is consistent with our
numerics, which interpolate all the way to strong coupling.
