We analyze the size evolution of HII regions around 27 quasars between z = 5.7 to 6.4 ( ’ quasar near-zones ’ or NZ ) . We include more sources than previous studies , and we use more accurate redshifts for the host galaxies , with 8 CO molecular line redshifts and 9 MgII redshifts . We confirm the trend for an increase in NZ size with decreasing redshift , with the luminosity normalized proper size evolving as : R _ { NZ,corrected } = ( 7.4 \pm 0.3 ) - ( 8.0 \pm 1.1 ) \times ( 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 \sim 9.4 from z = 6.4 to 5.7 , in its simplest interpretation . Alternatively , recent numerical simulations suggest that this rapid increase in near-zone 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 \sim 6 to 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 near-zone sizes increase with quasar UV luminosity , as expected for photo-ionization dominated by quasar radiation .