In Politis (2002) a method of tail index estimation for heavy-tailed time series, based on examining the growth rate of the logged sample second moment of the data, was proposed and studied. This estimator has a slow rate of convergence to the tail index, which is due to the high dependence of the summands of the statistic. To ameliorate the convergence rate, this work proposes an estimator with reduced bias, computed over subblocks of the whole data set. The resulting estimator obtains a polynomial rate of consistency for the tail index, and in simulation studies shows itself decidedly superior to competing prior art, such as the above-mentioned estimator of Politis (2002), as well as the reknowned Hill estimator.