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 $\beta \ge 0$ and $\beta <0$, but the lower bounds are not the sharp. In the paper, we give an improvement of the lower bounds for $\beta \ge 0$, i.e., $${\chi }_{\beta }(L(G))\ge \{\begin{array}{rlr}& {\chi }_{\beta }(G),& if\delta (G)\le 2,\\ & 2(1+\frac{2}{\mathrm{\Delta }+3}{)}^{\beta }{\chi }_{\beta }(G),& if\delta (G)\ge 3,\end{array}$$ and characterize the extremal graphs. In addition, for $\beta <0$, we present a small improvement on two special cases.