For any small constant $ε>0$, the Erdős-Rényi random graph $G(n,\frac{1+ε}{n})$ with high probability has a unique largest component which contains $(1\pm O(ε))2εn$ vertices. Let $G_c(n,p)$ be obtained by assigning each edge in $G(n,p)$ a color in $[c]$ independently and uniformly. Cooley, Do, Erde, and Missethan proved that for any fixed $α>0$, $G_{αn}(n,\frac{1+ε}{n})$ with high probability contains a rainbow tree (a tree that does not repeat colors) which covers $(1\pm O(ε))\fracα{α+1}εn$ vertices, and conjectured that there is one which covers $(1\pm O(ε))2εn$. In this paper, we achieve the correct leading constant and prove their conjecture correct up to a logarithmic factor in the error term, as we show that with high probability $G_{αn}(n,\frac{1+ε}{n})$ contains a rainbow tree which covers $(1\pm O(ε\log(1/ε)))2εn$ vertices.