  Recent numerical studies have revealed the physically intriguing fact that
charged black holes whose charge-to-mass ratios are larger than the critical
value $(Q/M)_{\text{crit}}=\sqrt{2(9+\sqrt{6})}/5$ can support hairy matter
configurations which are made of scalar fields with a non-minimal negative
coupling to the Gauss-Bonnet invariant of the curved spacetime. Using {\it
analytical} techniques, we explore the physical and mathematical properties of
the composed
charged-black-hole-nonminimally-coupled-linearized-massless-scalar-field
configurations in the near-critical $Q/M\gtrsim (Q/M)_{\text{crit}}$ regime. In
particular, we derive an analytical resonance formula that describes the
charge-dependence of the dimensionless coupling parameter
$\bar\eta_{\text{crit}}=\bar\eta_{\text{crit}}(Q/M)$ of the composed
Einstein-Maxwell-nonminimally-coupled-scalar-field system along the {\it
existence-line} of the theory, a critical border that separates bald
Reissner-Nordstr\"om black holes from hairy charged-black-hole-scalar-field
configurations. In addition, it is explicitly shown that the large-coupling
$-\bar\eta_{\text{crit}}(Q/M)\gg1$ analytical results derived in the present
paper for the composed Einstein-Maxwell-scalar theory agree remarkably well
with direct numerical computations of the corresponding black-hole-field
resonance spectrum.
