  We extend the analysis, initiated in arXiv:1901.01269, of the thermodynamic
stability of four-dimensional asymptotically flat hairy black holes by
considering a general class of exact solutions in Einstein-Maxwell-dilaton
theory with a non-trivial dilaton potential. We find that, regardless of the
values of the parameters of the theory, there always exists a sub-class of
hairy black holes that are thermodynamically stable and have the extremal limit
well defined. This generic feature that makes the equilibrium configurations
locally stable should be related to the properties of the dilaton potential
that is decaying towards the spatial infinity, but behaves as a box close to
the horizon. We prove that these thermodynamically stable solutions are also
dynamically stable under spherically symmetric perturbations.
