The spectra of high-redshift ($z\gtrsim 6$) quasars contain valuable information on the progression of the Epoch of Reionization (EoR). At redshifts $z<6$, the observed Lyman-series forest shows that the intergalactic medium (IGM) is nearly ionized, while at $z>7$ the observed quasar damping wings indicate high neutral gas fractions. However, there remains a gap in neutral gas fraction constraints at $6\lesssim z \lesssim 7$ where the Lyman series forest becomes saturated but damping wings have yet to fully emerge. In this work, we use a sample of 18 quasar spectra at redshifts $6.0Comment: 28 pages, 14 figures