We present an analysis of the evolution of the Lyman-series forest into the epoch of reionization using cosmological radiative transfer simulations in a scenario where reionization ends late. We explore models with different mid-points of reionization and gas temperatures. We find that once the simulations have been calibrated to match the mean flux of the observed Lyman-α (Ly α) forest at 4 < z < 6, they also naturally reproduce the distribution of effective optical depths of the Lyman-β (Ly β) forest in this redshift range. We note that the tail of the largest optical depths that is most challenging to match corresponds to the long absorption trough of ULAS J0148+0600, which we have previously shown to be rare in our simulations. We consider the evolution of the Lyman-series forest out to higher redshifts, and show that future observations of the Ly β forest at z > 6 will discriminate between different reionization histories. The evolution of the Ly α and Ly γ forests are less promising as a tool for pushing studies of reionization to higher redshifts due to the stronger saturation and foreground contamination, respectively.