  We discuss spontaneous scalarization of the Reissner-Nordstr\"om black hole
in the presence of higher derivative gauge field corrections that arise in the
context of string, as well as higher-dimensional more fundamental gravity
theories. Our theory admits the Reissner-Nordstr\"om solution at the scalar
vacuum of the theory ($\phi=0$) and we find that the higher order derivative
gauge field correction term results in the tachyonic instability of our system
once the coupling function satisfies the condition that its second derivative
is positive at the scalar vacuum in the appropriate parameter space. We find
that the branches do not end with an extremal black hole, rather with a
singularity as indicated by the divergence of the Kretschmann scalar. The black
holes can be overcharged in the sense that they may carry larger electric
charge in comparison to their mass. Finally, these solutions possess larger
entropy at the event horizon radius when compared to the Reissner-Nordstr\"om
black hole, as well as to scalarized black holes without the higher order
derivative gauge field terms, indicating in this way the thermodynamic
prefer-ability of our system, when compared to existing literature, while they
respect the energy conditions.
