  Bennett, Carbery and Tao considered the $k$-linear restriction estimate in
$\mathbb{R}^{n+1}$ and established the near optimal $L^\frac2{k-1}$ estimate
under transversality assumptions only. We have shown that the trilinear
restriction estimate improves its range of exponents under some curvature
assumptions. In this paper we establish almost sharp multilinear estimates for
a class of hypersurfaces with curvature for $4 \leq k \leq n$. Together with
previous results in the literature, this shows that curvature improves the
range of exponents in the multilinear restriction estimate at all levels of
lower multilinearity, that is when $k \leq n$.
