We report new measurements of the intergalactic medium (IGM) Ly$alpha$ and Ly$beta$ effective optical depth at $5.3<z<6.5$, using a new sample of quasar sightlines including 32 quasars at $6.308le zle7.00$. These quasars provide a large statistical sample to measure the IGM evolution during the transition phase of the reionization epoch. We construct a data set of deep optical spectra of these quasars using VLT, Keck, Gemini, LBT, and MMT. We measure the Ly$alpha$ effective optical depth at $5.36<z<6.57$ using the Ly$alpha$ forests of both individual spectra and the stacked spectrum. The large scatter of individual measurements is consistent with previous work, suggesting an inhomogeneous reionization process. Combining our new measurements and previous results, we obtain a best-fit for the Ly$alpha$ effective optical depth evolution at $z>5.3$, $taupropto(1+z)^{8.6pm1.0}$. We then estimate the observed Ly$beta$ effective optical depth using Ly$beta$ forests and convert them to Ly$alpha$ optical depth for comparison, which provides additional constraints on the evolution of the IGM optical depth. The Ly$beta$-based measurements are generally in agreement with the best-fit evolution obtained from Ly$alpha$ forests. Using this new sample, we identify 389 Ly$alpha$ and 50 Ly$beta$ transmission spikes at $5.5<z<6.3$. The upper limits of Ly$alpha$ optical depth estimated using transmission spikes are well consistent with our best-fit evolution. The evolution in number density of these high-redshift transmission spikes suggests a rapid transition phase at the end of the reionization. Comparison of our optical depth measurements with hydrodynamical simulations indicates a IGM neutral hydrogen fraction $langle f_{rm HI}ranglegtrsim10^{-4}$ at $z=6$.
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