  A possible cause of the late-time cosmic acceleration is an exotic fluid with
an equation of state lying within the phantom regime, i.e., $w=p/\rho <-1$. The
latter violates the null energy condition, which is a fundamental ingredient in
wormhole physics. Thus, cosmic phantom energy may, in principle, provide a
natural fluid to support wormholes. In this work, we find new asymptotically
flat wormhole solutions supported by the phantom energy equation of state,
consequently extending previous solutions. Thus, there is no need to surgically
paste the interior wormhole geometry to an exterior vacuum spacetime. In the
first example, we carefully construct a specific shape function, where the
energy density and pressures vanish at large distances as $\sim 1/r^{n}$, with
$n>0$. We also consider the "volume integral quantifier", which provides useful
information regarding the total amount of energy condition violating matter,
and show that, in principle, it is possible to construct asymptotically flat
wormhole solutions with an arbitrary small amount of energy condition violating
matter. In the second example, we analyse two equations of state, i.e.,
$p_r=p_r(\rho)$ and $p_t=p_t(\rho)$, where we consider a specific integrability
condition in order to obtain exact asymptotically flat wormhole solutions. In
the final example, we postulate a smooth energy density profile, possessing a
maximum at the throat and vanishing at spatial infinity.
