  It has been previously shown that a Gauss-Bonnet term non-minimally coupled
to a scalar field produces a scalarised black hole solution, which can be
considered as having secondary scalar hair, parametrised in terms of the black
hole's mass and charge. In this paper we extend a previously investigated
linear coupling of the form $f(\phi)=\phi$ to a non-minimally coupled Maxwell
term, with the form $\frac{1}{8}F_{\mu\nu}F^{\mu\nu}+\beta\phi
F_{\mu\nu}F^{\mu\nu}$. By using numerical methods the solutions to the full
differential equations are found, as well as a perturbative expansion in the
$r\rightarrow\infty$ limit and a perturbative expansion in couplings parameters
such as $\beta$. These solutions describe scalarised black holes with modified
electric field which have dependence not only on the electric charge of the
black hole, but also the value of the non-minimal coupling constant and the
horizon radius or mass of the black hole. We also discuss the bounds imposed on
the parameters of the black hole by the reality condition of the solution,
giving some explicit numerical bounds.
