The universality of the critical equation of state and the third law of thermodynamics require the specific entropy (per baryon) along the coexistence (first-order transition) line to be nonmonotonic at least on one side of that line. Specifically, a maximum must occur. We investigate the consequences of this simple observation for the isentropic trajectories near the QCD critical point. We identify a discriminant parameter, namely the sign of the isochoric derivative of pressure at the critical point, which determines on which side of the co-existence curve the non-monotonicity shall occur. We classify different scenarios of adiabatic expansion, relevant for heavy-ion collisions that arise depending on the value of the discriminant parameter and the proximity of the trajectory to the critical point.