Recent elaborated by T. Harko and collaborators, the f (R, T ) theories of gravity contemplate an optimistic alternative to dark energy, for which R and T stand for the Ricci scalar and the trace of the energymomentum tensor, respectively. Although the literature has shown that the T dependence on the gravitational part of the action -which is due to the consideration of quantum effects -may induce some novel features in the scope of late-time cosmological dynamics, in the radiation-dominated universe, when T = 0, no contributions seem to rise from such theories. Apparently, f (R, T ) contributions to a radiation-dominated universe may rise only from the f (R, T φ ) approach, which is nothing but the f (R, T ) gravity in the case of a self-interacting scalar field whose trace of the energymomentum tensor is T φ . We intend, in this article, to show how f (R, T φ ) theories of gravity can contribute to the study of the primordial stages of the universe. Our results predict a graceful exit from inflationary stage to a radiation-dominated era. They also predict a late-time cosmic acceleration after a matter-dominated phase, making the f (R, T φ ) theories able to describe, in a self-consistent way, all the different stages of the universe dynamics.