  The existence of global smooth solutions to the Navier-Stokes equations
(NSEs) with hyperviscosity $(-\Delta)^{\gamma}$ is open unless $\gamma $ is
close to the J.-L. Lions exponent $ \frac{5}{4}$ at which the energy balance is
strong enough to prevent singularity formation. If $1<\gamma \ll \frac{5}{4}$,
then the global well-posedness of the hyperviscous NSEs is widely open as for
the usual NSEs. In this paper, for all $\gamma>1$, we show the existence of a
transport noise for which global smooth solutions to the stochastic
hyperviscous NSEs on the three-dimensional torus exist with high probability.
In particular, a suitable transport noise considerably improves the known
well-posedness results in the deterministic setting.
