Abstract: We study the efficient estimation of predictive confidence intervals for black-box predictors when the common data exchangeability (e.g., i.i.d.) assumption is violated due to potentially feedback-induced shifts in the input data distribution. That is, we focus on standard and feedback covariate shift (FCS), where the latter allows for feedback dependencies between train and test data that occur in many decision-making scenarios like experimental design. Whereas prior conformal prediction methods for this problem are in general either extremely computationally demanding or make inefficient use of labeled data, we propose a collection of methods based on the jackknife+ that achieve a practical balance of computational and statistical efficiency. Theoretically, our proposed JAW-FCS method extends the rigorous, finite-sample coverage guarantee of the jackknife+ to FCS. We moreover propose two tunable relaxations to JAW-FCS's computation that maintain finite-sample guarantees: one using only $K$ leave-one-out models (JAW-$K$LOO) and a second building on $K$-fold cross validation+ (WCV+). Practically, we demonstrate that JAW-FCS and its computational relaxations outperform state-of-the-art baselines on a variety of real-world datasets under standard and feedback covariate shift, including for biomolecular design and active learning tasks.