A path in a graph $G$ is called non-self-touching if two vertices are neighbours in the path if and only if they are neighbours in the graph. We investigate the existence of doubly infinite non-self-touching paths in infinite plane graphs. The matching graph $G_*$ of an infinite plane graph $G$ is obtained by adding all diagonals to all faces, and it plays an important role in the theory of site percolation on $G$. The main result of this paper is a necessary and sufficient condition on $G$ for the existence of a doubly infinite non-self-touching path in $G_*$ that traverses some diagonal. This is a key step in proving, for quasi-transitive $G$, that the critical points of site percolation on $G$ and $G_*$ satisfy the strict inequality $p_c(G_*) < p_c(G)$, and it complements the earlier result of Grimmett and Li (Random Struct. Alg. 65 (2024) 832--856), proved by different methods, concerning the case of transitive graphs. Furthermore it implies, for quasi-transitive graphs, that $p_u(G) + p_c(G) \ge 1$, with equality if and only if the graph $G_Δ$, obtained from $G$ by emptying all separating triangles, is a triangulation. Here, $p_u$ is the critical probability for the existence of a unique infinite open cluster.