In this paper, we show that every $O(m)$-edge-connected simple graph $G$ of size divisible by $m$ with minimum degree at least $2^{O(m)}$ has an edge-decomposition into isomorphic copies of any given tree $T$ of size $m$. Moreover, the minimum degree condition can be dropped for graphs $G$ with girth greater than the diameter of $T$. These results improve two results due to Bensmail, Harutyunyan, Le, Merker, and Thomassé (2017) and Merker (2017) who gave a factorial upper bound on the necessary edge-connectivity.