  We revisit the problem of the small Bjorken-$x$ asymptotics of the quark and
gluon orbital angular momentum (OAM) distributions in the proton utilizing the
revised formalism for small-$x$ helicity evolution derived recently in a paper
by Cougoulic, Kovchegov, Tarasov, and Tawabutr. We relate the quark and gluon
OAM distributions at small $x$ to the polarized dipole amplitudes and their
(first) impact-parameter moments. To obtain the $x$-dependence of the OAM
distributions, we derive novel small-$x$ evolution equations for the
impact-parameter moments of the polarized dipole amplitudes in the
double-logarithmic approximation (summing powers of $\alpha_s \ln^2(1/x)$ with
$\alpha_s$ the strong coupling constant). We solve these evolution equations
numerically and extract the large-$N_c$, small-$x$ asymptotics of the quark and
gluon OAM distributions, which we determine to be \[
  L_{q+\bar{q}}(x, Q^2) \sim L_{G}(x,Q^2) \sim \Delta \Sigma(x, Q^2) \sim
\Delta G(x,Q^2) \sim \left(\frac{1}{x}\right)^{3.66 \, \sqrt{\frac{\alpha_s
N_c}{2\pi}}},\] in agreement with an earlier work by Boussarie, Hatta, and Yuan
within the precision of our numerical evaluation (here $N_c$ is the number of
quark colors). We also investigate the ratios of the quark and gluon OAM
distributions to their helicity distribution counterparts in the small-$x$
region.
