  We calculate the spectrum of three-body Efimov bound states near a Feshbach
resonance within a model which accounts both for the finite range of
interactions and the presence of background scattering. The latter may be due
to direct interactions in an open channel or a second overlapping Feshbach
resonance. It is found that background scattering gives rise to substantial
changes in the trimer spectrum as a function of the detuning away from a
Feshbach resonance, in particular in the regime where the background channel
supports Efimov states on its own. Compared to the situation with negligible
background scattering, the regime where van der Waals universality applies is
shifted to larger values of the resonance strength if the background scattering
length is positive. For negative background scattering lengths, in turn, van
der Waals universality extends to even small values of the resonance strength
parameter, consistent with experimental results on Efimov states in $^{39}$K.
Within a simple model, we show that short-range three-body forces do not affect
van der Waals universality significantly. Repulsive three-body forces may,
however, explain the observed variation between around $-8$ and $-10$ of the
ratio between the scattering length where the first Efimov trimer appears and
the van der Waals length.
