Summary: The edge weight, denoted by \(\Delta_e(G)\), of a graph \(G\) is max \(\{d_G(u)+d_G(v):uv\in E(G)\}\). An edge-coloring of a graph \(G\) is injective if for any two distinct edges \(e_1\) and \(e_2\), the colors of \(e_1\) and \(e_2\) are distinct if they are at distance 2 in \(G\) or in a common triangle. The injective chromatic index of \(G\), denoted by \(\chi_i^\prime(G)\), is the minimum number of colors needed for an injective edge-coloring of \(G\). B. Ferdjallah et al. [“Injective edge-coloring of sparse graphs”, Preprint, arXiv:1907.09838] posed a conjecture which says that if \(G\) is subcubic graph, then \(\chi_i^\prime(G) \le 6\). A. Kostochka et al. [Eur. J. Comb. 96, Article ID 103355, 12 p. (2021; Zbl 1466.05072)] showed that if \(G\) is a subcubic graph, then \(\chi_i^\prime(G) \le 7\). We extend the above result by showing that \[ \chi^\prime_i(G)\le \begin{cases} 4, & \quad{\text{ if } \Delta_e(G)\le 5,}\\ 7, & \quad{\text{ if } \Delta_e(G)\le 6,}\\ 12, & \quad{\text{ if } \Delta_e(G)\le 7}. \end{cases} \] In addition, we prove that \(\chi_i^\prime(G) \le 6\) for any subcubic claw-free graph, and this bound is sharp.