  We compute the sphaleron rate in quenched QCD for a temperature $T \simeq
1.24~T_c$ from the inversion of the Euclidean lattice time correlator of the
topological charge density. We explore and compare two different strategies:
one follows a new approach proposed in this study and consists in extracting
the rate from finite lattice spacing correlators, and then in taking the
continuum limit at fixed smoothing radius followed by a zero-smoothing
extrapolation; the other follows the traditional approach of extracting the
rate after performing such double extrapolation directly on the correlator. In
both cases the rate is obtained from a recently-proposed modification of the
standard Backus-Gilbert procedure. The two strategies lead to compatible
estimates within errors, which are then compared to previous results in the
literature at the same or similar temperatures; the new strategy permits to
obtain improved results, in terms of statistical and systematic uncertainties.
