  The component of the neoclassical electrostatic potential that is
non-constant on the magnetic surface, that we denote by $\tilde\varphi$, can
affect radial transport of highly charged impurities, and this has motivated
its inclusion in some modern neoclassical codes. The number of neoclassical
simulations in which $\tilde\varphi$ is calculated is still scarce, partly
because they are usually demanding in terms of computational resources,
especially at low collisionality. In this paper the size, the scaling with
collisionality and with aspect ratio, and the structure of $\tilde\varphi$ on
the magnetic surface are analytically derived in the $1/\nu$, $\sqrt{\nu}$ and
superbanana-plateau regimes of stellarators close to omnigeneity; i. e.
stellarators that have been optimized for neoclassical transport. It is found
that the largest $\tilde\varphi$ that the neoclassical equations admit scales
linearly with the inverse aspect ratio and with the size of the deviation from
omnigeneity. Using a model for a perturbed omnigeneous configuration, the
analytical results are verified and illustrated with calculations by the code
KNOSOS. The techniques, results and numerical tools employed in this paper can
be applied to neoclassical transport problems in tokamaks with broken
axisymmetry.
