  We give constraints on the existence of $(\chi,b)$-factors in the global
$A$-parameter of a genuine cuspidal automorphic representation $\sigma$ of a
metaplectic group in terms of the invariant, lowest occurrence index, of theta
lifts to odd orthogonal groups. We also give a refined result that relates the
invariant, first occurrence index, to non-vanishing of period integrals of
residues of Eisenstein series associated to the cuspidal datum
$\chi\otimes\sigma$. This complements our previous results for symplectic
groups.
