We present all the partial wave descriptions of the nucleon–nucleus system by proposing a new additive phenomenological potential with emphasis on off-energy-shell scattering. For most of the general treatment of the physical processes, the off-shell transition matrices are most expedient quantities because they carry as much information as the potential. As the off-shell Jost solution is an indispensable ingredient for deriving transition matrices, we initially construct this function by taking into account the ordinary differential equation method. Finally,we execute certain tests on our expressions with respect to various limiting conditions and present numerical results using the $MATLAB$ programme. Numerical results are in sensible conformity with the previous works.