  We investigate extremal dyon black holes in the Einstein-Maxwell-dilaton
(EMD) theory with higher curvature corrections in the form of the Gauss-Bonnet
density coupled to the dilaton. In the same theory without the Gauss-Bonnet
term the extremal dyon solutions exist only for discrete values of the dilaton
coupling constant $a$. We show that the Gauss-Bonnet term acts as a dyon hair
tonic enlarging the allowed values of $a$ to continuous domains in the plane
$(a, q_m)$ the second parameter being the magnetic charge. In the limit of the
vanishing curvature coupling (a large magnetic charge) the dyon solutions
obtained tend to the Reissner-Nordstr\"om solution but not to the extremal
dyons of the EMD theory. Both solutions have the same values of the horizon
radius as a function of charges. The entropy of new dyonic black holes
interpolates between the Bekenstein-Hawking value in the limit of the large
magnetic charge (equivalent to the vanishing Gauss-Bonnet coupling) and twice
this value for the vanishing magnetic charge. Although an expression for the
entropy can be obtained analytically using purely local near-horizon solutions,
its interpretation as the black hole entropy is legitimate only once the global
black hole solution is known to exist, and we obtain numerically the
corresponding conditions on the parameters. Thus, a purely local analysis is
insufficient to fully understand the entropy of the curvature corrected black
holes. We also find dyon solutions which are not asymptotically flat, but
approach the linear dilaton background at infinity. They describe magnetic
black holes on the electric linear dilaton background.
