We obtain upper bounds for the rates of convergence for the simple random walk Green's function in the domains $D_α= D_α(n)=\{re^{iθ}\in \mathbb{C}:0 <θ<2π-α, 0<r<2n\}-z_0,$ where $z_0\in\mathbb{Z}^2$ is a point closest to $ne^{i(π-α/2)}$. The rate depends on the angle of the wedge and is what was suggested by the sharpest available results in the extreme cases $α=0$ and $α=π$. Our proof uses the KMT coupling between random walk and Brownian motion.