  We continue our study of the cubic fractional NLS with very weak dispersion
$\alpha>1$ and data distributed according to the Gibbs measure. We construct
the natural strong solutions for
$\alpha>\alpha_0=\frac{31-\sqrt{233}}{14}\approx 1.124$ which is strictly
smaller than $\frac{8}{7}$, the threshold beyond which the first nontrivial
Picard iteration has no longer the Sobolev regularity needed for the
deterministic well-posedness theory. This also improves our previous result in
Sun-Tzvetkov \cite{Sun-Tz2}. We rely on recent ideas of Bringmann
\cite{Bringmann} and Deng-Nahmod-Yue \cite{Deng2}. In particular we adapt to
our situation the new resolution ansatz in \cite{Deng2} which captures the most
singular frequency interaction parts in the $X^{s,b}$ type space. To overcome
the difficulties caused by the weakly dispersive effect, our specific strategy
is to benefit from the "almost" transport effect of these singular parts and to
exploit their $L^{\infty}$ as well as the Fourier-Lebesgue property in order to
inherit the random feature from the linear evolution of high frequency
portions.
