  The phenomenon of spontaneous scalarisation of charged black holes (BHs) has
recently motivated studies of various Einstein-Maxwell-scalar models. Within
these models, different classes of BH solutions are possible, depending on the
non-minimal coupling function $f(\phi)$, between the scalar field and the
Maxwell invariant. Here we consider the class wherein both the (bald)
electrovacuum Reissner-Nordstr\"om (RN) BH and new scalarised BHs co-exist, and
the former are never unstable against scalar perturbations. In particular we
examine the model, within this subclass, with a quartic coupling function:
$f(\Phi) = 1+\alpha \Phi ^4$. The domain of existence of the scalarised BHs,
for fixed $\alpha$, is composed of two branches. The first branch (cold
scalarised BHs) is continuously connected to the extremal RN BH. The second
branch (hot scalarised BHs) connects to the first one at the minimum value of
the charge to mass ratio and it includes overcharged BHs. We then assess the
perturbative stability of the scalarised solutions, focusing on spherical
perturbations. On the one hand, cold scalarised BHs are shown to be unstable by
explicitly computing growing modes. The instability is quenched at both
endpoints of the first branch. On the other hand, hot scalarised BHs are shown
to be stable by using the S-deformation method. Thus, in the spherical sector
this model possesses two stable BH local ground states (RN and hot scalarised).
We point out that the branch structure of BHs in this model parallels the one
of BHs in five dimensional vacuum gravity, with [Myer-Perry BHs, fat rings,
thin rings] playing the role of [RN, cold scalarised, hot scalarised] BHs.
