In this paper, we examine a class of $α$-potential stochastic differential games with random coefficients via the backward stochastic differential equations (BSDEs) approach. Specifically, we show that the first and second order linear derivatives of the objective function for each player can be expressed through the corresponding first and second-order adjoint equations, which leads to rigorous estimates for $α$. We illustrate the dependence of $α$ on game characteristics through detailed analysis of linear-quadratic games, and with common noise.