  In a previous paper [Phys. Rev. D 97, 104044 (2018)] we pointed out some
shortcomings of the standard approach to chameleon theories consisting in
treating the small bodies used to test the weak equivalence principle (WEP) as
test particles, whose presence do not modify the chameleon field configuration.
In that paper we developed an alternative method to determine the relevant
field configuration which takes into account the influence of both test and
source bodies, and computed the chamaleon mediated force. Relying on that
analysis we showed that the effective acceleration of test bodies is
composition dependent even when the model is based on universal couplings. In
this paper, we improve our method by using a more suitable approximation for
the effective chameleon potential in situations where the bodies are in the
so-called "thick shell regime". We then find new and more restrictive bounds on
the model' s parametres by confronting the new theoretical predictions with the
empirical bounds on E\"otv\"os parameter comming from the lunar laser ranging
experiments.
