Within the scope of a FRW cosmological model we have studied the role of spinor field in the evolution of the Universe when it is non-minimally coupled to the gravitational one. We considered a few types of nonlinearity. It was found that if the spinor field nonlinearity describes an ordinary matter such as radiation, the presence of non-minimality becomes essential and leads to the rapid expansion of the Universe, whereas if the spinor field nonlinearity describes a dark energy, the evolution of the Universe is dominated by it and the difference between the minimal and non-minimally coupled cases becomes almost indistinguishable. Hence it should be argued that non-minimal coupling can be viewed as an alternative way to describe the accelerated mode of expansion of the Universe. © 2020, Springer Nature B.V.