  We analyze the world polarized deep-inelastic scattering (DIS) and
semi-inclusive DIS (SIDIS) data at low values of $x < 0.1$, using small-$x$
evolution equations for the flavor singlet and nonsinglet helicity parton
distribution functions (hPDFs). The hPDFs for quarks, antiquarks, and gluons
are extracted and evolved to lower values of $x$ to make predictions for the
future Electron-Ion Collider (EIC). We improve on our earlier work by employing
the more realistic large-$N_c\, \& N_f$ limit of the revised small-$x$ helicity
evolution, and incorporating running coupling corrections along with SIDIS data
into the fit. We find an anti-correlation between the signs of the gluon and
$C$-even quark hPDFs as well as the $g_1$ structure function. While the
existing low-$x$ polarized DIS and SIDIS data are insufficient to constrain the
initial conditions for the polarized dipole amplitudes in the helicity
evolution equations, future EIC data will allow more precise predictions for
hPDFs and the $g_1$ structure function for $x$ values beyond those probed at
the EIC. Using the obtained hPDFs, we discuss the contributions to the proton
spin from quark and gluon spins at small $x$.
