Drawbacks of traditional approximate (Wald test-based) and exact (Clopper-Pearson) confidence intervals for a binomial proportion are well-recognized. Alternatives include an interval based on inverting the score test, adaptations of exact testing, and Bayesian credible intervals derived from uniform or Jeffreys beta priors. We recommend a new interval intermediate between the Clopper-Pearson and Jeffreys in terms of both width and coverage. Our strategy selects a value κ between 0 and 0.5 based on stipulated coverage criteria over a grid of regions comprising the parameter space, and bases lower and upper limits of a credible interval on Beta(κ, 1- κ) and Beta(1- κ, κ) priors, respectively. The result tends toward the Jeffreys interval if the criterion is to ensure an average overall coverage rate (1-α) across a single region of width 1, and toward the Clopper-Pearson if the goal is to constrain both lower and upper lack of coverage rates at α/2 with region widths approaching zero. We suggest an intermediate target that ensures all average lower and upper lack of coverage rates over a specified set of regions are ≤ α/2. Interval width subject to these criteria is readily optimized computationally, and we demonstrate particular benefits in terms of coverage balance.