Bayes factors (BFs) are becoming increasingly important tools in genetic association studies, partly because they provide a natural framework for including prior information. The Wakefield BF (WBF) approximation is easy to calculate and assumes a normal prior on the log odds ratio (logOR) with a mean of zero. However, the prior variance (W) must be specified. Because of the potentially high sensitivity of the WBF to the choice of W, we propose several new BF approximations with , but allow W to take a probability distribution rather than a fixed value. We provide several prior distributions for W which lead to BFs that can be calculated easily in freely available software packages. These priors allow a wide range of densities for W and provide considerable flexibility. We examine some properties of the priors and BFs and show how to determine the most appropriate prior based on elicited quantiles of the prior odds ratio (OR). We show by simulation that our novel BFs have superior true-positive rates at low false-positive rates compared to those from both P-value and WBF analyses across a range of sample sizes and ORs. We give an example of utilizing our BFs to fine-map the CASP8 region using genotype data on approximately 46,000 breast cancer case and 43,000 healthy control samples from the Collaborative Oncological Gene-environment Study (COGS) Consortium, and compare the single-nucleotide polymorphism ranks to those obtained using WBFs and P-values from univariate logistic regression.