  The small-$x$ quark helicity evolution equations at double-logarithmic order,
with the kernel $\sim\alpha_s\ln^2(1/x)$, have been derived previously. In this
work, we derive the single-logarithmic corrections to the equations, to order
$\alpha_s\ln(1/x)$ of the evolution kernel. The new equations include the
effects of the running coupling and the unpolarized small-$x$ evolution, both
of which are parametrically significant at single-logarithmic order. The
large-$N_c$ and large-$N_c\& N_f$ approximations to the equation are computed.
(Here, $N_c$ and $N_f$ are the numbers of quark colors and flavors,
respectively.) Their solutions will provide more precise estimates of the quark
helicity distribution at small $x$, contributing to the resolution of the
proton spin puzzle.

High Energy Physics - Phenomenology;High Energy Physics - Experiment;Nuclear Experiment;Nuclear Theory