A unified multiscalar field model with three flat regions is discussed. The three flat regions are the inflation, early- and late dark energy epochs. The potential is obtained by a spontaneous breaking of scale invariance generated by non-Riemannian measures of integration (or two measures theories (TMT)). We define the scale invariant couplings of the scalar fields to the different measures through exponential potentials. Spontaneous breaking of scale invariance takes place when integrating the fields that define the measures. When going to the Einstein frame we obtain: (i) An effective potential for the scalar fields with three flat regions which allows for a unified description of both early Universe inflation (in the higher-energy density flat region) as well as of present dark energy epoch which can be realized with a double phase, i.e., in two flat regions. (ii) In the slow-roll inflation, only one field combination, the "dilaton,"which transforms under scale transformations, has nontrivial dynamics; the orthogonal one, which is scale invariant, remains constant. The corresponding perturbations of the dilaton are calculated. (iii) For a reasonable choice of the parameters the present model perturbations conforms to the Planck Collaboration data. (iv) In the late Universe we define scale-invariant couplings of Dark Matter to the dilaton. These couplings define a matter-induced potential for the dilaton and extremizing this potential determines the scale-invariant scalar field, while all exotic noncanonical behavior of the Dark Matter as well as any possible fifth force disappear. (v) We calculate the evolution of the late Universe under these conditions with the realization of two different possible realizations of Λ cold dark matter-type scenarios depending on the flat region in the late Universe. These two phases could appear at different times in the history of the Universe. (vi) From the Planck data, we find the constraints on the parameters during the inflationary epoch and these values are used to obtain constraints relevant to the present epoch.