  We develop a method for calculating the doublet splittings of select
ground-state single $\Lambda$-hypernuclei. This hypernuclear spectroscopy is
conducted by supplementing the self-consistent single-particle equations with
an effective interaction, which follows directly from the underlying
lagrangian,to simulate the residual particle-hole interaction. Our previous
investigation, performed using only the leading-order contributions to the
particle-hole interaction, was inadequate. In the present work, this method is
improved upon by increasing the level of truncation in the residual interaction
to include gradient coupling to the neutral vector meson, and thereby
incorporating the tensor force into the calculation (which is known to play a
crucial role in these systems). As a result, we obtain a realistic description
of the effect of the tensor coupling on the doublet orderings and splittings.
