  We consider the most general CP-conserving renormalizable effective scalar
potential involving two doublets plus one singlet Higgs and satisfying the
electroweak gauge symmetry. After deriving the electroweak-symmetry breaking
conditions, we focus on special cases, characterized by specific symmetry
properties and/or relations to supersymmetry-inspired extensions of the
Standard Model (e.g. n/NMSSM, UMSSM). We then investigate the question of the
reconstruction of the potential parameters from the Higgs masses and mixing
angles and show that in some specific cases, such as the one of an underlying
NMSSM, an accuracy at the order of leading-logarithms is achievable with
minimal effort. We finally study a few phenomenological consequences for this
latter model. More specifically, we consider how our parameter reconstruction
modifies the outcome of two publicly available codes : micrOMEGAs and
NMSSMTools. We observed noteworthy effects in regions of parameter space where
Higgs-to-Higgs decays are relevant, impacting the collider searches for light
Higgs states and the prediction of the Dark-Matter relic density.
