This paper presents correction terms to the tangent approximation for the first-exit density of Brownian motion at distant boundaries. These lead to second-order approximations to the first-exit distribution. Asymptotic formulas for the mean and variance of the first-exit time are derived. Numerical comparisons show the accuracy of the approximations.
Christel Jennen. "Second-Order Approximations to the Density, Mean and Variance of Brownian First-Exit Times." Ann. Probab. 13 (1) 126 - 144, February, 1985. https://doi.org/10.1214/aop/1176993071