  We study the transverse single-spin asymmetry with a $\sin\phi_S$ modulation
in semi-inclusive deep inelastic scattering. Particularly, we consider the case
in which the transverse momentum of the final state hadron is integrated out.
Thus, the asymmetry is merely contributed by the coupling of the transversity
distribution function $h_1(x)$ and the twist-3 collinear fragmentation function
$\tilde{H}(z)$. Using the available parametrization of $h_1(x)$ from SIDIS data
and the recent extracted result for $\tilde{H}(z)$, we predict the $\sin\phi_S$
asymmetry for charged and neutral pion production at an Electron Ion Collider.
We find that the asymmetry is sizable and could be measured. We also study the
impact of the leading-order QCD evolution effect and find that it affects the
$\sin\phi_S$ asymmetry at EIC considerably.
