Intracortical brain-computer interfaces (iBCIs) require frequent recalibration to maintain robust performance due to changes in neural activity that accumulate over time. Compensating for this nonstationarity would enable consistently high performance without the need for supervised recalibration periods, where users cannot engage in free use of their device. Here we introduce a hidden Markov model (HMM) to infer what targets users are moving toward during iBCI use. We then retrain the system using these inferred targets, enabling unsupervised adaptation to changing neural activity. Our approach outperforms the state of the art in large-scale, closed-loop simulations over two months and in closed-loop with a human iBCI user over one month. Leveraging an offline dataset spanning five years of iBCI recordings, we further show how recently proposed data distribution-matching approaches to recalibration fail over long time scales; only target-inference methods appear capable of enabling long-term unsupervised recalibration. Our results demonstrate how task structure can be used to bootstrap a noisy decoder into a highly-performant one, thereby overcoming one of the major barriers to clinically translating BCIs.