  As is well known, the smallest neutrino mass turns out to be vanishing in the
minimal seesaw model, since the effective neutrino mass matrix $M^{}_\nu$ is of
rank two due to the fact that only two heavy right-handed neutrinos are
introduced. In this paper, we point out that the one-loop matching condition
for the effective dimension-five neutrino mass operator can make an important
contribution to the smallest neutrino mass. By using the available one-loop
matching condition and two-loop renormalization group equations in the
supersymmetric version of the minimal seesaw model, we explicitly calculate the
smallest neutrino mass in the case of normal neutrino mass ordering and find
$m^{}_1 \in [10^{-10}, 10^{-8}]~{\rm eV}$ at the Fermi scale $\Lambda^{}_{\rm
F} = 91.2~{\rm GeV}$, where the range of $m^{}_1$ results from the
uncertainties on the choice of the seesaw scale $\Lambda^{}_{\rm SS}$ and on
the input values of relevant parameters at $\Lambda^{}_{\rm SS}$.
