  In GUT-scale constrained (GUTc) supersymmetric (SUSY) models, the mass of
smuon $\tilde{\mu}_1$ is typically heavier than that of stau $\tilde{\tau}_1$,
and stau co-annihilation is a typical annihilation mechanism of dark matter.
However, light smuon is more favored by the muon $g-2$ anomaly, thus
smuon-neutralino loop contribution to muon $g-2$ is usually smaller than that
of sneutrino-chargino. Inspired by the latest muon $g-2$ results, we take the
GUTc- Next-to-Minimal Supersymmetric Model (NMSSM) as an example, where the
gaugino (Higgs) masses are not unified to the usual parameter $M_{1/2}$
($M_0$), exploring its possibility of light smuon and its contribution to muon
$g-2$. After complicated calculations and discussions, we conclude that in
GUTc-NMSSM the smuon can be lighter than stau. In this light-smuon scenario,
the contribution of smuon-neutralino loop to the muon $g-2$ can be larger than
that of the sneutrino-chargino loop. The annihilation mechanisms of dark matter
are dominated by multiple slepton or chargino co-annihilation. In our
calculations, we consider also other latest related constraints like Higgs
data, SUSY searches, dark matter relic density and direct detections, etc.
