  We consider charged black holes with scalar hair obtained in a class of
Einstein-Maxwell-scalar models, where the scalar field is coupled to the
Maxwell invariant with a quartic coupling function. Besides the
Reissner-Nordstr\"om black holes, these models allow for black holes with
scalar hair. Scrutinizing the domain of existence of these hairy black holes,
we observe a critical behavior: A limiting configuration is encountered at a
critical value of the charge, where spacetime splits into two parts: an inner
spacetime with a finite scalar field and an outer extremal Reissner-Nordstr\"om
spacetime. Such a pattern was first observed in the context of gravitating
non-Abelian magnetic monopoles and their hairy black holes.
