  We consider a relaxation of the concept of well-covered graphs, which are
graphs with all maximal independent sets of the same size. The extent to which
a graph fails to be well-covered can be measured by its independence gap,
defined as the difference between the maximum and minimum sizes of a maximal
independent set in $G$. While the well-covered graphs are exactly the graphs of
independence gap zero, we investigate in this paper graphs of independence gap
one, which we also call almost well-covered graphs. Previous works due to
Finbow et al. (1994) and Barbosa et al. (2013) have implications for the
structure of almost well-covered graphs of girth at least $k$ for $k\in
\{7,8\}$. We focus on almost well-covered graphs of girth at least $6$. We show
that every graph in this class has at most two vertices each of which is
adjacent to exactly $2$ leaves. We give efficiently testable characterizations
of almost well-covered graphs of girth at least $6$ having exactly one or
exactly two such vertices. Building on these results, we develop a
polynomial-time recognition algorithm of almost well-covered
$\{C_3,C_4,C_5,C_7\}$-free graphs.
