  This is a companion paper to earlier work of the authors, which proved an
integral surgery formula for framed instanton homology. First, we present an
enhancement of the large surgery formula, a rational surgery formula for
null-homologous knots in any 3-manifold, and a formula encoding a large portion
of $I^\sharp(S^3_0(K))$. Second, we use the integral surgery formula to study
the framed instanton homology of many 3-manifolds: Seifert fibered spaces with
nonzero orbifold degrees, especially nontrivial circle bundles over any
orientable surface, surgeries on a family of alternating knots and all twisted
Whitehead doubles, and splicings with twist knots. Finally, we use the previous
techniques and computations to study almost L-space knots, ${\it i.e.}$, the
knots $K\subset S^3$ with $\dim I^\sharp(S_n^3(K))=n+2$ for some
$n\in\mathbb{N}_+$. We show that an almost L-space knot of genus at least $2$
is fibered and strongly quasi-positive, and a genus-one almost L-space knot
must be either the figure eight or the mirror of the $5_2$ knot in Rolfsen's
knot table.
