We revisit the the quark and gluon orbital angular momentum (OAM) distributions in the proton at small $x$. Utilizing the revised small-$x$ helicity evolution from an earlier study, we calculate the quark and gluon OAM distributions at small $x$, and relate them to the polarized dipole amplitudes and their (first) impact-parameter moments. To obtain the small-$x$ asymptotics 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
$$\begin{align} \notag L_{q+\bar{q}}(x, Q^2) \sim L_{G}(x,Q^2) \sim \Delta \Sigma(x, Q^2) \sim \Delta G(x,Q^2) \sim \left(\frac{1}{x}\right)^{3.66 \, \sqrt{\frac{\alpha_s N_c}{2\pi}}}, \end{align}$$
in agreement with earlier results in the literature, 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.