For a real number \( \beta \), the general sum-connectivity index \( \chi_{\beta}(G) \) of a graph \( G \) is defined as \( \chi_{\beta}(G)=\sum_{xy\in E(G)}(d_G(x)+d_G(y))^{\beta} \), where \( d(x) \) denote the degree of a vertex \( x \) in \( G \). In Chen (2023), the author present the lower bounds for \( \chi_\beta(L(G)) \) in terms of \( \chi_\beta(G) \) for \( \beta\geq 0 \) and \( \beta < 0 \), but the lower bounds are not the sharp. In the paper, we give an improvement of the lower bounds for \( \beta\geq 0 \), i.e., \[ \chi_\beta(L(G))\geq\left\{\begin{aligned} &\chi_\beta(G),&if~\delta(G)\leq 2,\\ &2(1+\frac{2}{\Delta+3})^\beta \chi_\beta(G),&if~\delta(G)\geq 3, \end{aligned}\right. \] and characterize the extremal graphs. In addition, for \( \beta< 0 \), we present a small improvement on two special cases.