  We apply the formalism developed earlier for studying transverse momentum
dependent parton distribution functions (TMDs) at small Bjorken $x$ to
construct the small-$x$ asymptotics of the quark Sivers function. First, we
explicitly construct the complete fundamental "polarized Wilson line" operator
to sub-sub-eikonal order: this object can be used to study a variety of quark
TMDs at small-$x$. We then express the quark Sivers function in terms of dipole
scattering amplitudes containing various components of the "polarized Wilson
line" and show that the dominant (eikonal) term which contributes to the quark
Sivers function at small $x$ is the spin-dependent odderon, confirming the
recent results of Dong, Zheng and Zhou. Our conclusion is also similar to the
case of the gluon Sivers function derived by Boer, Echevarria, Mulders and Zhou
(see also the work by Szymanowski and Zhou). We also analyze the sub-eikonal
corrections to the quark Sivers function using the constructed "polarized
Wilson line" operator. We derive new small-$x$ evolution equations re-summing
double-logarithmic powers of $\alpha_s \, \ln^2 (1/x)$ with $\alpha_s$ the
strong coupling constant. We solve the corresponding novel evolution equations
in the large-$N_c$ limit, obtaining a sub-eikonal correction to the
spin-dependent odderon contribution. We conclude that the quark Sivers function
at small $x$ receives contributions from two terms and is given by
\begin{align} f_{1 \: T}^{\perp \: q} (x, k_T^2) = C_O (x, k_T^2) \,
\frac{1}{x} + C_1 (k_T^2) \, \left( \frac{1}{x} \right)^0 + \ldots \end{align}
with the function $C_O (x, k_T^2)$ varying slowly with $x$ and the ellipsis
denoting the sub-asymptotic and sub-sub-eikonal (order-$x$) corrections.
