  Higher-order derivative terms are considered as replacement for the Skyrme
term in an Einstein-Skyrme-like model in order to pinpoint which properties are
necessary for a black hole to possess stable static scalar hair. We find two
new models able to support stable black hole hair in the limit of the Skyrme
term being turned off. They contain 8 and 12 derivatives, respectively, and are
roughly the Skyrme-term squared and the so-called BPS-Skyrme-term squared. In
the twelfth-order model we find that the lower branches, which are normally
unstable, become stable in the limit where the Skyrme term is turned off. We
check this claim with a linear stability analysis. Finally, we find for a
certain range of the gravitational coupling and horizon radius, that the
twelfth-order model contains 4 solutions as opposed to 2. More surprisingly,
the lowest part of the would-be unstable branch turns out to be the stable one
of the 4 solutions.
