  Within the nuclear shell model, we investigate the correction $\delta_{RO}$
to the Fermi matrix element due to a mismatch between proton and neutron
single-particle radial wave functions. Eight superallowed $0^+ \to 0^+$ $\beta$
decays in the $sd$-shell, comprised of $^{22}$Mg, $^{26m}$Al, $^{26}$Si,
$^{30}$S, $^{34}$Cl, $^{34}$Ar, $^{38m}$K and $^{38}$Ca are re-examined. The
radial wave functions are obtained from a spherical Woods-Saxon potential whose
parametrizations are optimized in a consistent adjustment of the depth and the
length parameter to relevant experimental observables, such as nucleon
separation energies and charge radii, respectively. The chosen fit strategy
eliminates the strong dependence of the radial mismatch correction to a
specific parametrization, except for calculations with an additional
surface-peaked term. As an improvement, our model proposes a new way to
calculate the charge radii, based on a parentage expansion which accounts for
correlations beyond the extreme independent-particle model. Apart from the
calculations with a surface-peak term and the cases where we used a different
model space, the new sets of $\delta_{RO}$ are in general agreement with the
earlier result of Towner and Hardy [1]. Small differences of the corrected
average $\overline{\mathcal{F}t}$ value are observed.
