  The minimal length of closed geodesics among all finite-type hyperbolic
surfaces with self-intersection number $k$ is $2\cosh^{-1}(2k+1)$ for $k$
sufficiently large, i.e. $k>10^{13350}$ by arXiv:2210.12966. In the present
paper we will give an another proof independent of it and improve the lower
bound $k>10^{13350}$ to $k>1750$.
