Sponsored search ranking requires accurate and well-calibrated estimates of click and purchase probabilities. But these predictive tasks remain challenging because click and sale labels are sparse, and clicked-only post-click conversion modeling introduces sample selection bias. In this work, we exploit the simple but useful observation that for small probabilities, the logit of the probability is approximately equal to the log of the probability, which yields an additive logit-space decomposition of the purchase prediction task into a click-propensity term and a learned residual. We then introduce a simple, model agnostic, modular approach for overall p(sale) prediction models by directly injecting $p(click)$ model outputs into a downstream $p(sale)$ model as a feature, a prior, or both, giving a flexible, additive alternative to the usual multiplicative click--conversion factorization for sparse purchase prediction. Training both click and sale models on the entire space of impression data mitigates sample selection bias while allowing the sale model to learn nonlinear interactions of click propensity with post-click behavior, rather than relying on a rigid multiplicative decomposition. This provides a practical and modular path to incorporate denser click signals into the sparse purchase prediction task, across many modeling architectures. Offline results at a major e-commerce company showed consistent gains in classification and calibration performance metrics; online A/B tests gave statistically significant gains in user and platform metrics; and a variant of the method was launched to production.