Consider the following general question: if we can solve Maximum Matching in (quasi) linear time on a graph class C, does the same hold true for the class of graphs that can be modularly decomposed into C ? What makes the latter question difficult is that the Maximum Matching problem is not preserved by quotient, thereby making difficult to exploit the structural properties of the quotient subgraphs of the modular decomposition. So far, we are only aware of a recent framework in (Coudert et al., SODA'18) that only applies when the quotient subgraphs have bounded order and/or under additional assumptions on the nontrivial modules in the graph. As an attempt to answer this question for distance-hereditary graphs and some other superclasses of cographs, we study the combined effect of modular decomposition with a pruning process over the quotient subgraphs. Specifically, we remove sequentially from all such subgraphs their so-called one-vertex extensions (i.e., pendant, anti-pendant, twin, universal and isolated vertices). Doing so, we obtain a "pruned modular decomposition", that can be computed in quasi linear time. Our main result is that if all the pruned quotient subgraphs have bounded order then a maximum matching can be computed in linear time. The latter result strictly extends the framework of Coudert et al. Our work is the first to use some ordering over the modules of a graph, instead of just over its vertices, in order to speed up the computation of maximum matchings on some graph classes.