In this work, exact analytical expressions for regular solution and on-shell Jost function are calculated for nuclear Hulthén plus atomic Hulthén potential by imposing the same range approximation to both nuclear andatomic potentials. In this context, the regular solution is utilised to find expressions for the off-shell Jost function and half-shell T -matrix. The half-off-shell T -matrix and the elastic scattering phase shifts for the nucleon–nucleon and nucleus–nucleus systems are computed. Our results are found to be in good agreement with the standard data.