  We investigate the initial value problem (IVP) associated to the modified
Korteweg-de Vries equation (mKdV) in the defocusing scenario: \begin{equation*}
\left\{\begin{array}{l} \partial_t u+ \partial_x^3u-u^2\partial_x(u) = 0, \quad
x,t\in\mathbb{R}, \\ u(x,0) = u_0(x), \end{array}\right. \end{equation*} where
$u$ is a real valued function and the initial data $u_0$ is analytic on
$\mathbb{R}$ and has uniform radius of analyticity $\sigma_0$ in the spatial
variable.
  It is well-known that the solution $u$ preserves its analyticity with the
same radius $\sigma_0$ for at least some time span $0<T_0\le 1$. This local
result was obtained in [Nonlinear Differ. Equ. Appl. (2024), 31--68] by proving
a trilinear estimate in the Gevrey spaces $G^{\sigma, s}$, $s\geq \frac14$.
Global in time behaviour of the solution and algebraic lower bound of the
evolution of the radius of analyticity was also studied in authors' earlier
works in [Nonlinear Differ. Equ. Appl. (2024), 31--68] and [J. Evol. Equ. 24
No. 42 (2024)] by constructing almost conserved quantities in the classical
Gevrey space with $H^1$ and $H^2$ levels of Sobolev regularities. The present
study aims to construct a new almost conservation law in the Gevrey space
defined with a weight function $\cosh(\sigma|\xi|)$ and use it demonstrate that
the local solution $u$ extends globally in time, and the radius of spatial
analyticity is bounded from below by $c T^{-\frac{1}{2}}$, for any time $T\geq
T_0$. The outcome of this paper represents an improvement on the one achieved
by the authors' previous work in [J. Evol. Equ. 24 No. 42 (2024)].
