We study Keisler measures in strongly n-distal NIP theories, generalizing some results of Simon and Chernikov-Starchenko for distal theories and addressing some questions of Walker. In particular, we establish a hypergraph version of the distal regularity lemma, compact domination for definable fsg groups, and demonstrate that the strong n-distality hierarchy is strict among stable theories using a connection to Poizat's total triviality of forking. We also show that infinite strongly n-distal NIP fields have characteristic 0 using a discrepancy result of Babai-Hayes-Kimmel from multiparty communication complexity.