  Jang has proven a remarkable classification of $6$-dimensional manifolds
having an almost complex circle action with $4$ fixed points. Jang classifies
the weights and associated multigraph into six cases, leaving the existence of
connected manifolds fitting into three of the cases unknown. We show that one
of the unknown cases may be constructed by a surgery construction of Kustarev,
and the underlying manifold is diffeomorphic to $S^4 \times S^2$. We show that
the action is not equivariantly diffeomorphic to a linear one, thus giving a
new exotic $S^1$-action of on a product of spheres that preserves an almost
complex structure. We also prove a uniqueness statement for the almost complex
structures produced by Kustarev's construction and prove some topological
applications of Jang's classification.
