We demonstrate that a certain dimensional family of symmetric singly-periodic minimal surface with Scherk-type ends exists in the neighborhood of a given example if a set of holomorphic quadratic differentials is independent. Our approach extends the work of Traizet, who has previously shown the existence of a family of such minimal surfaces in the neighborhood of a degenerate example consisting of a number of intersecting planes. Whereas in Traizet's construction the underlying conformal structure was a union of Riemann spheres, we treat the case where the underlying conformal structure is a Riemann surface of higher genus. In our approach, admissible surfaces are identified with Weierstrass data satisfying certain constraints. Using the bilinear relations and the Rauch variational formula, we are able to find holomorphic quadratic differentials which represent differentials of the constraints, and whose independence, by an implicit function theorem argument, implies the existence of the desired surface family in a neighborhood of the original. We restrict our attention to tori and develop machinery for investigating the quadratic differentials numerically using interval arithmetic to obtain provable bounds on their residue structure. The developed tools are finally applied to an example surface in Karcher's one-dimensional toroidal saddle tower family, which is shown to exist in a larger three-dimensional family.