The inhomogeneous cosmological reionization process leaves tangible imprints in the intergalactic medium (IGM) down to z ~ 4-5. The Lyα forest flux power spectrum provides a potentially powerful probe of the epoch of reionization. With the existing Sloan Digital Sky Survey I/II quasar sample, we show that two cosmological reionization scenarios, one completing reionization at z = 6 and the other at z = 9, can be distinguished at ~7σ level by utilizing Lyα forest absorption spectra at z = 3.9-4.1 in the absence of other physical processes that may also affect the Lyα flux power spectrum. The difference may not be distinguishable at such high significance after marginalization over other effects, but, in any case, one will need to consider this effect in order to correctly interpret the power spectrum in this redshift range. The redshift range z = 4-5 may provide the best window because there are still enough transmitted flux and quasars to measure precise statistics of the flux fluctuations, and the IGM still retains a significant amount of memory of reionization.