In this paper we consider the problem of computing Seshadri constants at a general point on a smooth polarized toric surface. We consider the case when the degree of jet separation is small or the core of the associated polygon is a line segment. Our main result is that under these hypothesis the Seshadri constant at the general point can often be determined in terms of easily computable invariants of the surfaces at hand. When the core of the associated polygon is a point we show that the surface can be constructed via consecutive equivariant blow-ups of either P2 or P1× P1.