The post here: Solving the min edge cover using the maximum matching algorithm provides a way to obtain the min edge cover from a maximum matching by greedily adding edges on top of the maximum matching until all vertices are covered. Now, thinking about the min-weighted edge cover problem, it would seem this approach can be extended. First, find the minimum weighted matching with maximum edges, and then add edges of the smallest weight greedily, smallest weight ones first.
However, reading section 19.3 of the book "Combinatorial optimization: polyhedra and efficiency" by Schrijver, a more complicated algorithm is presented. This makes it seem like my approach above is sub-optimal. Is it possible to find a counter-example, preferably on a bi-partite graph where my greedy algorithm would fail to provide an optimal solution? I haven't been able to find one with some toy graphs.