We investigate traveling solitary waves in the one-dimensional (1D) Salerno model, which interpolates between the cubic discrete nonlinear Schrodinger (DNLS) equation and the integrable Ablowitz-Ladik (AL) model. In a traveling frame the model becomes an advance-delay differential equation to which we analyze the existence of homoclinic orbits to the rest state. The method of beyond all orders asymptotics is used to compute the so-called Stokes constant that measures the splitting of the stable and unstable manifolds. Through computing zeros of the Stokes constant, we identify a number of solution families that may bifurcate for parameter values between the DNLS and AL limits of the Salerno model. Using a pseudospectral method, we numerically continue these solution families and show that their parameters approach the curves of the zero level of the Stokes constant as the soliton amplitude approaches zero. An interesting topological structure of solutions occurs in parameter space. As the AL limit is approached, solution sheets of single-hump solutions undergo folds and become double-hump solitons. Numerical simulation suggests that the single-humps are stable and interact almost inelastically.