We analyze the size evolution of H II regions around 27 quasars between z = 5.7 and 6.4 ('quasar near zones' or NZs). We include more sources than previous studies, and we use more accurate redshifts for the host galaxies, with eight CO molecular line redshifts and nine Mg II redshifts. We confirm the trend for an increase in NZ size with decreasing redshift, with the luminosity-normalized proper size evolving as R{sub NZ,corrected} = (7.4 {+-} 0.3) - (8.0 {+-} 1.1) x (z - 6) Mpc. While derivation of the absolute neutral fraction remains difficult with this technique, the evolution of the NZ sizes suggests a decrease in the neutral fraction of intergalactic hydrogen by a factor {approx}9.4 from z = 6.4 to 5.7, in its simplest interpretation. Alternatively, recent numerical simulations suggest that this rapid increase in NZ size from z = 6.4 to 5.7 is due to the rapid increase in the background photo-ionization rate at the end of the percolation or overlap phase, when the average mean-free path of ionizing photons increases dramatically. In either case, the results are consistent with the idea that z {approx} 6-7 corresponds to the tail end of cosmic reionization. The scatter in the normalized NZ sizes is larger than expected simply from measurement errors, and likely reflects intrinsic differences in the quasars or their environments. We find that the NZ sizes increase with quasar UV luminosity, as expected for photo-ionization dominated by quasar radiation.