The locally rotationally symmetric Bianchi type-I cosmological models are explored both in general relativity and in an alternative modified theory of gravity, viz., f(R, T) gravity. Solutions have been found by assuming a special form for the Hubble parameter, which results in a hyperbolic hybrid scale factor. The geometrical features of the model, including the jerk parameter, have been discussed. In order to ensure the viability of the solutions, various constraints have been imposed. A comparison is made between the solutions in general relativity and in f(R, T) gravity. It is found that both the solutions in general relativity and in f(R, T) gravity possess very rich behaviour. In general relativity and in f(R, T) gravity, the equation of state permits a transition from normal (stiff) matter to quintessence, phantom, and then later to a solution that mimics the cosmological constant, depending on the values of m, n, Q and lambda  (the various parameters of the solutions). Our results are illustrated throughout by means of relevant diagrams. It is found that in general, the solutions exhibit late-time acceleration both in general relativity as well as in f(R, T) gravity.