  We improve our previous result [L. Molinet and T. Tanaka, Unconditional
well-posedness for some nonlinear periodic one-dimensional dispersive
equations, J. Funct. Anal. 283 (2022), 109490] on the Cauchy problem for one
dimensional dispersive equations with a quite general nonlinearity in the
periodic setting. Under the same hypotheses that the dispersive operator
behaves for high frequencies as a Fourier multiplier by $ i |\xi|^\alpha \xi $,
with $ 1 \le \alpha\le 2 $, and that the nonlinear term is of the form $
\partial_x f(u) $ where $f $ is a real analytic function whose Taylor series
around the origin has an infinite radius of convergence, we prove the
unconditional LWP of the Cauchy problem in $H^s(\mathbb{T}) $ for $ s\ge
1-\frac{\alpha}{4} $ with $ s>1/2 $. It is worth noticing that this result is
optimal in the case $\alpha=2$ (generalized KdV equation) in view of the
restriction $ s>1/2 $ for the continuous injection of $ H^s(\mathbb{T}) $ into
$ L^\infty(\mathbb{T}) $. Our main new ingredient is the remplacement of
improved Strichartz estimates by improved bilinear estimates in the treatment
of the worst resonant interactions. Such improved bilinear estimates already
appeared in the work of Hani in the context of Schr\"odinger equations on a
compact manifold. Finally note that this enables us to derive global existence
results for $ \alpha \in [4/3,2] $.
