  This paper discusses wormholes supported by general equation-of-state ,
resulting in a significant combination of the linear equation-of-state and some
other models. Wormhole with a quadratic equation-of-state is studied as a
particular example. It is shown that the violation of null energy condition is
restricted to some regions in the vicinity of the throat. The combination of
barotropic and polytropic equation-of-state has been studied. We consider fluid
near the wormhole throat in an exotic regime which at some $r=r_{1}$, the
exotic regime is connected to a distribution of asymptotically dark energy
regime with $-1<\omega<-1/3$. We have presented wormhole solutions with small
amount of exotic matter. We have shown that using different forms of
equation-of-state has a considerable effect on the minimizing violation of the
null energy condition. The effect of many parameters such as redshift as
detected by a distant observer and energy density at the throat on the $r_1$ is
investigated. The solutions are asymptotically flat and compatible with
presently available observational data at the large cosmic scale.
