The mean free path of ionizing photons, λ mfp, is a critical parameter for modeling the intergalactic medium (IGM) both during and after reionization. We present direct measurements of λ mfp from QSO spectra over the redshift range 5 < z < 6, including the first measurements at z ≃ 5.3 and 5.6. Our sample includes data from the XQR-30 VLT large program, as well as new Keck/ESI observations of QSOs near z ∼ 5.5, for which we also acquire new [C ii] 158 μm redshifts with ALMA. By measuring the Lyman continuum transmission profile in stacked QSO spectra, we find , , , and pMpc at z = 5.08, 5.31, 5.65, and 5.93, respectively. Our results demonstrate that λ mfp increases steadily and rapidly with time over 5 < z < 6. Notably, we find that λ mfp deviates significantly from predictions based on a fully ionized and relaxed IGM as late as z = 5.3. By comparing our results to model predictions and indirect λ mfp constraints based on IGM Lyα opacity, we find that the evolution of λ mfp is consistent with scenarios wherein the IGM is still undergoing reionization and/or retains large fluctuations in the ionizing UV background well below redshift 6.