I am not too experienced on this topic, so take my answer with a grain of salt.
PCM (Polarizable Continuum Model) deals with solvation effects by assuming that the effect of solvent molecules surrounding the solute cavity can be treated by adding point charges on the cavity surface (generally determined by considering some kind of scaled van der Waals radii). Then the solute can be polarized by the solvent field, which can in turn be polarized by the solute—and this can be solved by an iterative procedure to get the final polarized wavefunction of the solute.
However, the assumption in this method is that the solute charge is entirely constrained inside the cavity (Because there is only one layer of point charges outside of the solute, which is supposed to represent all of the solvent surrounding it). But when the calculations are allowed to run without any constraint, the wavefunction of the solute always penetrates outside the solvation cavity. This is usually referred to as "escaped charges". This leads to extra polarization of the solute, as you mention in the question.
A way to model the actual solute-solvent interaction is to add more layers of point charges on the solute, to model the layers of solvent around the solute. (until most of the wavefunction of the solute is inside the layers of solvent.) This method is called the surface and volume polarization for electrostatic interaction(SVPE).
The good thing about this method is that it is possible to add more and more solvent layers, until the problem of escaped charges becomes negligible, at which point, SVPE solvation gives the exact surface and volume polarization which would be obtained if the Poisson's equations were analytically solved.
There is also SS(V)PE, which attempts to simulate the SVPE solvation at low cost by putting some extra point charges instead of modelling each layer explicitly.
SVPE or SS(V)PE gives the electrostatic part of the solvent effect, and the non-electrostatic part can be added to it to get more accurate solvation free energies. (Just like in PCM, adding CDS terms gives the SMD model). There are two models—one where parameter-fitted non-electrostatic corrections were taken directly from SMD (called SMVLE), and other is where parameter-fitted dispersion, exchange and short-range extremum(DEFESR) correction is added(called CMIRS).
These methods give very good results, especially with ions. For example, CMIRS v1.0 gives a mean unsigned error in hydration energy of 2.4 kcal/mol against experimental data for ions, whereas SMD gives 4 kcal/mol error with ions. The data is not reported separately for anions and cations, but anions in general show higher errors in solvation energies (J. Phys. Chem. A 2019, 123, 44, 9498–9504).
CMIRS v1.1 (newer parameterization) is implemented in GAMESS and Q-Chem. SMVLE is implemented only in GAMESS afaik.
- M. J. Vilkas, C. G. Zhan, "An efficient implementation for determining volume polarization in self-consistent reaction field theory", J. Chem. Phys., 2008, 129(19), 194109.