  Wormholes with Ellis geometry have been successfully constructed using
tachyon matter \cite{Das}. However, for such a wormhole, it is obtained that
the redshift function is necessarily a constant, and also the wormhole is
plagued with an imaginary tachyon potential and a constant field if the
solutions are obtained in the absence of a cosmological constant term. So, a
physically plausible wormhole solution is possible only in the presence of a
$\Lambda$ term. In this paper, we try to construct a wormhole from tachyon
matter with three $other$ geometries \textit{different} from the Ellis geometry
and see whether it is possible to construct them successfully, besides checking
whether the restrictions of the Ellis wormhole can be overcome with these
geometries. Among others, we obtain one very interesting result that for all
three of these geometries \textit{different} from the Ellis, the $\Lambda$ term
is no longer an essential ingredient in constructing physically plausible
traversable wormholes and the tachyon matter, capable of providing explanations
for the "\textbf{dark sector}" of the universe,is itself sufficient for this
purpose.
