Abstract Let F be a family of connected graphs. A graph G is said to be F-free if G is H-free for every graph H in F. Fujita et al. (2006)  studied the problem of characterizing the families of graphs F such that every large enough connected F-free graph of odd order has a near perfect matching. In the same work, the authors characterized such families F, but in the case where every graph in F is triangle-free. In this paper, we complete the characterization of all such families, removing the triangle-free condition.