  We re-examine our previous calculation on the Collins effect in
semi-inclusive deeply inelastic scattering (SIDIS) process with a $^3$He
target, and find that our previous treatment on the dilution factors may cause
the results larger than the realistic situation. We thus modify our calculation
in an improved treatment with an updated prediction on the
$\sin(\phi_h+\phi_S)$ asymmetry for the JLab 12 GeV under the transverse
momentum dependent (TMD) factorization framework. Meanwhile, we also provide
the prediction of such asymmetry for the JLab 6 GeV and the prediction of the
$\sin(3\phi_h-\phi_S)$ asymmetry related to pretzelosity.
