  We extend our analysis of transverse single spin asymmetry (SSA) in
electroproduction of $J/\psi$ to include the effect of the scale evolution of
the transverse momentum dependent (TMD) parton distribution functions and gluon
Sivers function. We estimate SSA for JLab, HERMES, COMPASS, and eRHIC energies
using the color evaporation model of charmonium production, using an
analytically obtained approximate solution of TMD evolution equations discussed
in the literature. We find that there is a reduction in the asymmetry compared
with our predictions for the earlier case considered by us, wherein the $Q^2$
dependence came only from DGLAP evolution of the unpolarized gluon densities
and a different parametrization of the TMD Sivers function was used.
