  In previous work, it has been shown that the recently proposed LLR method is
very efficient at overcoming strong metastabilities that arise near first-order
phase transition points. Here we present a systematic study of the performance
of the algorithm near (pseudo-)critical points for $q$-state Potts models with
$q$ as large as 20, in two and three dimensions. In particular, we shall focus
our study on the ergodicity of the replica exchange step and the underlying
physical mechanism. When compared with both analytical and numerical results
present in the literature, our determinations of thermodynamic observables
(including the order-disorder interface tension at criticality) show an
impressive degree of relative accuracy (up to $2.5 \times 10^{-6}$), which
confirms the reliability and the efficiency of the proposed approach.
