  The classical theta correspondence, based on the Weil representation, allows
one to lift automorphic representations on symplectic groups or their double
covers to automorphic representations on special orthogonal groups. It is of
interest to vary the orthogonal group and describe the behavior in this theta
tower (the Rallis tower). In prior work, the authors obtained an extension of
the classical theta correspondence to higher degree metaplectic covers of
symplectic and special orthogonal groups that is based on the tensor product of
the Weil representation with another small representation. In this work we
study the existence of generic lifts in the resulting theta tower. In the
classical case, there are two orthogonal groups that may support a generic lift
of an irreducible cuspidal automorphic representation of a symplectic group. We
show that in general the Whittaker range consists of $r+1$ groups for the lift
from the $r$-fold cover of a symplectic group. We also give a period criterion
for the genericity of the lift at each step of the tower.
