Gauss-Bonnet-dilatonic coupling in four dimensions plays an important role to explain late-time cosmic evolution. However, this term is an outcome of the low energy string effective action and thus ought to be important in the early universe too. Unfortunately, a phase-space formulation of such a theory does not exist in the literature due to branching. We therefore consider a modified theory of gravity, which contains a nonminimally coupled scalar-tensor sector in addition to a higher-order scalar curvature invariant term with Gauss-Bonnet-dilatonic coupling. Such an action unifies early inflation with late-time cosmic acceleration. The quantum version of the theory is also well behaved.