  In this paper we study the Gevrey smoothing effect of solutions to the
non-cutoff spatially homogeneous and inhomogeneous Boltzmann equation for soft
potential. We consider the mild singularity case $s<1/2$ as we did in the
previous work for spatially homogeneous case (J. Diff. Equ. 253(4) (2012),
1172-1190. DOI: 10.1016/j.jde.2012.04.023) and for spatially inhomogeneous case
(arXiv:1304.2971), and try to extend the range of $\gamma$. We derive a new
coercivity estimate for collision operator, using which we can obtain the
Gevrey regularity for $\gamma \in (-5/2,0)$ improving the previous assumption
$\gamma \in (-1-2s,0)$. Besides, we consider $\gamma$ and $s$ separately
instead of viewing $\gamma+2s$ as one untied quantity.
