  We consider the matching augmentation problem (MAP), where a matching of a
graph needs to be extended into a $2$-edge-connected spanning subgraph by
adding the minimum number of edges to it. We present a polynomial-time
algorithm with an approximation ratio of $13/8 = 1.625$ improving upon an
earlier $5/3$-approximation. The improvement builds on a new
$\alpha$-approximation preserving reduction for any $\alpha\geq 3/2$ from
arbitrary MAP instances to well-structured instances that do not contain
certain forbidden structures like parallel edges, small separators, and
contractible subgraphs. We further introduce, as key ingredients, the technique
of repeated simultaneous contractions and provide improved lower bounds for
instances that cannot be contracted.
