  We obtain the local well-posedness of the one dimensional cubic nonlinear
Schr\"odinger Equation for initial data in the modulation space $M_{2, p}$ for
all $2\le p<\infty$, which covers all the subcritical cases from the viewpoint
of scaling. Moreover, in order to approach the endpoint space $M_{2,\infty}$,
we will prove the almost global well-posedness in some Orlicz-type space, which
is a natural generalisation of $M_{2,p}$ for large $p$. The new ingredient is
an endpoint version of the two dimensional restriction estimate.
