- We consider a non-standard generalized model of gravity coupled to a neutral scalar "inflaton" as well as to the fields of the electroweak bosonic sector. The essential new ingredient is employing two alternative non-Riemannian space-time volume-forms (non-Riemannian volume elements, or covariant integration measure densitities) independent of the space-time metric. The latter are defined in terms of auxiliary antisymmentric tensor gauge fields, which although not introducing any additional propagating degrees of freedom, trigger a series of important features such as: (i) appearance of two infinitely large flat regions of the effective "inflaton" potential in the corresponding Einstein frame with vastly different scales corresponding to the "early" and "late" epochs of Universe's evolution; (ii) dynamical generation of Higgs-like spontaneous symmetry breaking effective potential for the SU(2)\times U(1) iso-doublet electroweak scalar in the late universe, whereas it remains massless in the "early" universe. Next, to stabilize the quintessential dynamics, we introduce in addition a coupling of the "inflaton" to Gauss-Bonnet gravitational term. The latter leads to the following radical change of the form of the total effective "inflaton" potential: its flat regions are now converted into a local maximum corresponding to a "hill-top" inflation in the "early" universe with no spontaneous breakdown of electroweak gauge symmetry and, correspondigly, into a local minimum corresponding to the "late" universe evolution with a very small value of the dark energy and with operating Higgs mechanism.