  For most aspherical Seifert-fibered 3-manifolds $M$, the space of Seifert
fiberings $SF(M)$ is known to have contractible components. It is also known
that the space of Hopf fiberings of the three-sphere is noncontractible. We
provide the second example of a non-aspherical 3-manifold $M$ such that $SF(M)$
has noncontractible components. In particular, we show that certain components
of $SF(S^1 \times S^2)$ are homotopy equivalent to a subspace homeomorphic to
the identity-based loop space $\Omega SO(3)$, and we exhibit second homology
generators for both connected components of $SF(S^1 \times S^2)$.
