  Let the circle act in a Hamiltonian fashion on a connected compact symplectic
manifold $(M, \omega)$ of dimension $2n$. Then the $S^1$-action has at least
$n+1$ fixed points. In a previous paper, we study the case when the fixed point
set consists of precisely $n+1$ isolated points. In this paper, we study the
case when the fixed point set consists of exactly $n+2$ isolated points. We
show that in this case $n$ must be even. We find equivalent conditions on the
first Chern class of $M$ and a particular weight of the $S^1$-action. We also
show that the particular weight can completely determine the integral
cohomology ring and the total Chern class of $M$, and the sets of weights of
the $S^1$-action at all the fixed points. We will see that all these data are
isomorphic to those of known examples, $\widetilde{G}_2(\mathbb{R}^{n+2})$ with
$n\geq 2$ even, equipped with standard circle actions.
