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A complete understanding of parton helicity inside the proton requires the knowledge about its contribution at small Bjorken-$x$, which is difficult to deduce from experiments. We attempt to fill the gap by deriving a small-$x$ helicity evolution, resumming powers of $\alpha_s\ln^2(1/x)$, with $\alpha_s$ the strong coupling constant. Recently, a revised version of this evolution has been constructed, taking into account the observation that the evolution of the sub-eikonal operator, $\overleftarrow{D}^i\, D^i$, mixes with other helicity-dependent operators from the previous works, which are the gluon field strength, $F^{12}$, and the quark axial current, ${\bar \psi} \gamma^+ \gamma^5 \psi$. Based on the new evolution, a closed system of evolution equations can be constructed in the limits of large $N_c$ or large $N_c\& N_f$. (Here, $N_c$ and $N_f$ are the number of quark colors and flavors, respectively.) We numerically solve the equations in these limits and obtain the following small-$x$ asymptotics for the $g_1$ structure function at $N_f \leq 5$: $g_1(x,Q^2) \sim (1/x)^{\alpha_h \sqrt{\alpha_s N_c/2\pi}}$, with the intercept, $\alpha_h$, decreasing with $N_f$. In particular, at the large-$N_c$ limit, we have $\alpha_h = 3.66$, which agrees with the earlier work by Bartels, Ermolaev and Ryskin. Once the sixth quark flavor is turned on, i.e. $N_f = 6$, an oscillatory pattern in $\ln\frac{1}{x}$ emerges. However, the oscillation period spans many units of rapidity, making it difficult to observe in an experiment.