  An analysis of the $\alpha$ + core properties of $^{104}$Te along with a
global discussion on the $\alpha$-cluster structure above the double-shell
closures is presented from the viewpoint of the local potential model. The
$\alpha$ + core interaction is described by a nuclear potential of (1 +
Gaussian)$\times$(W.S. + W.S.$^3$) shape with two free parameters, which has
been successfully tested in nuclei of different mass regions. The model
produces $Q_{\alpha}$ values and $\alpha$-decay half-lives for $^{104}$Te in
agreement with the 2018 experimental data of Auranen et al. in the energy range
$5.13 \; \mathrm{MeV} < Q_{\alpha} < 5.3 \; \mathrm{MeV}$ using an $\alpha$
preformation factor $P_{\alpha} = 1$. The comparison of the calculated reduced
$\alpha$-widths for the ground state bands of $^{104}$Te, $^{94}$Mo and
$^{212}$Po indicates that $^{104}$Te has an $\alpha$-cluster degree
significantly higher than $^{94}$Mo and much higher than $^{212}$Po. These
results point to $^{104}$Te as a preferential nucleus for $\alpha$-clustering
in the $N,Z = 50$ region and corroborate the statement on the superallowed
$\alpha$-decay in $^{104}$Te.
