The $2$-Edge-Connected Spanning Subgraph problem (2-ECSS) is one of the most fundamental and well-studied problems in the context of network design. In the problem, we are given an undirected graph $G$, and the objective is to find a $2$-edge-connected spanning subgraph $H$ of $G$ with the minimum number of edges. For this problem, a lot of approximation algorithms have been proposed in the literature. In particular, very recently, Garg, Grandoni, and Ameli gave an approximation algorithm for 2-ECSS with factor $1.326$, which was the best approximation ratio. In this paper, we give a $(1.3+\varepsilon)$-approximation algorithm for 2-ECSS, where $\varepsilon$ is an arbitrary positive fixed constant, which improves the previously known best approximation ratio. In our algorithm, we compute a minimum triangle-free $2$-edge-cover in $G$ with the aid of the algorithm for finding a maximum triangle-free $2$-matching given by Hartvigsen. Then, with the obtained triangle-free $2$-edge-cover, we apply the arguments by Garg, Grandoni, and Ameli.