  Motivated by a novel origin of transverse single spin asymmetry (SSA) in
semi-inclusive Deep Inelastic Scattering (SIDIS) uncovered by some of us, we
quantitatively investigate its impact on the theoretical understanding of the
mechanism responsible for SSA. This new contribution from the quark-initiated
channel first appears in two-loop perturbation theory and involves the $g_T(x)$
distribution. We point out another entirely analogous piece from the
gluon-initiated channel proportional to the gluon helicity distribution $\Delta
G(x)$. Both contributions are solely expressed in terms of twist-two polarized
parton distribution functions and twist-two fragmentation functions in the
Wandzura-Wilczek approximation, such that they can be unambiguously evaluated
without introducing free parameters. We make predictions for measurements of
the asymmetries $A_{UT}$ at the future Electron-Ion Collider (EIC), and find
that $A_{UT}$ associated with the $\sin (\phi_h-\phi_S)$, $\sin \phi_S$ and
$\sin (2\phi_h-\phi_S)$ harmonics can reach up to 1-2\% even at the top EIC
energy.
