A classical problem in Distance Geometry, with multiple practical applications (in molecular structure determination, sensor network localization etc.) is to find the possible placements of the vertices of a graph with given edge lengths. For minimally rigid graphs, the double-exponential Gröbner Bases algorithm with an elimination order can be applied, in theory, but it is impractical even for small instances. By relating the problem to the computation of circuit polynomials in the Cayley-Menger ideal, we recently proposed an algebraic-combinatorial approach and an elimination algorithm for circuit polynomials [23]. It is guided by a tree structure whose leaves correspond to complete $K_4$ graphs and whose nodes perform algebraic resultant operations. In this paper we uncover further combinatorial structure in the Cayley-Menger algebraic matroid that leads to an extension of our algorithm. In particular, we generalize the combinatorial resultant operation of [23] to take advantage of the non-circuit generators and irreducible polynomials in the Cayley-Menger ideal and use them as leaves of the tree guiding the elimination. Our new method has been implemented in Mathematica and allows previously unobtainable calculations to be carried out. In particular, the $K_{3,3}$-plus-one circuit polynomial, with over one million terms in 10 variables and whose calculation crashed after several days with the previous method of [23], succeeded now in approx. 30 minutes.