For a quadratic endomorphism of the affine line defined over the rationals, we consider the problem of bounding the number of rational points that eventually land at the origin after iteration. In the article "Uniform Bounds on Pre-Images Under Quadratic Dynamical Systems", by the present authors and five others, it was shown that the number of rational iterated pre-images of the origin is bounded as one varies the morphism in a certain one-dimensional family. Away from a finite collection of morphisms in the family, we provide a sharp bound on the number of rational iterated pre-images. Then we give a conditional bound over the whole family using a number of modern tools for locating rational points on high genus curves. We also provide further insight into the geometry of the "pre-image curves".