In this paper, we present an improved union bound on the Linear Programming (LP) decoding performance of binary linear codes transmitted over an additive white Gaussian noise channel. The bounding technique is based on the Hunter bound, which is a second-order upper bound in probability theory, and it is minimized by Prim's minimum spanning tree algorithm. The bound calculation needs the fundamental cone generators of a given parity-check matrix rather than only their weight distribution, but involves relatively low computational complexity. It is targeted to high-density parity-check codes, where the number of their generators is extremely large and these generators are densely distributed in the Euclidean space. We explore the generator density and make a comparison between different parity-check matrix representations. That density affects the improvement of the proposed bound over the conventional LP union bound. This paper also presents a complete pseudo-weight distribution of the fundamental cone generators for the BCH[31,21,5] code.