We extend the position-space renormalization procedure, where renormalization factors are calculated from Greens functions in position space, by introducing a technique to take the average of Greens functions over spheres. In addition to reducing discretization errors, this technique enables the resulting position-space correlators to be evaluated at any physical distance, making them continuous functions similar to the $O(4)$-symmetric position-space Greens functions in the continuum theory but with a residual dependence on a regularization parameter, the lattice spacing $a$. We can then take the continuum limit of these renormalized quantities calculated at the same physical renormalization scale $|x|$ and investigate the resulting $|x|$-dependence to identify the appropriate renormalization window. As a numerical test of the spherical averaging technique, we determine the renormalized light and strange quark masses by renormalizing the scalar current. We see a substantial reduction of discretization effects on the scalar current correlator and an enhancement of the renormalization window. The numerical simulation is carried out with $2+1$-flavor domain-wall fermions at three lattice cutoffs in the range 1.79--3.15~GeV.
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