  In a previous paper it was shown that a one-turning-point WKB approximation
gives an accurate picture of the spectrum of certain non-Hermitian PT-symmetric
Hamiltonians on a finite interval with Dirichlet boundary conditions.
Potentials to which this analysis applies include the linear potential $V=igx$
and the sinusoidal potential $V=ig\sin(\alpha x)$. However, the
one-turning-point analysis fails to give the full structure of the spectrum for
the cubic potential $V=igx^3$, and in particular it fails to reproduce the
critical points at which two real eigenvalues merge and become a
complex-conjugate pair. The present paper extends the method to cases where the
WKB path goes through a {\it pair} of turning points. The extended method gives
an extremely accurate approximation to the spectrum of $V=igx^3$, and more
generally it works for potentials of the form $V=igx^{2N+1}$. When applied to
potentials with half-integral powers of $x$, the method again works well for
one sign of the coupling, namely that for which the turning points lie on the
first sheet in the lower-half plane.
