This paper investigates the proton kinetic mechanisms leading to the formation of plasma jets in antiparallel magnetic reconnection. In particular, the interaction of the protons with the Hall electric field in the proton non-ideal region is discussed. The study, based on a two-dimensional hybrid simulation, details the important role of the proton pressure force in the acceleration process and its role in maintaining open and steady the proton outflow channel. When no fluid closure is assumed, it is found that this force arises from a strong anisotropy in velocity space which comes from kinetic effect. By analyzing the distribution functions and the individual particle dynamics, it is shown that the mixing of protons bouncing in a divergent electrostatic potential well associated to the Hall effect statistically couples the two in-plane velocity components of the particles. This coupling results, from the macroscopic point of view, in off-diagonal components of the pressure tensor.