ONEAudit provides more efficient risk-limiting audits than other extant methods when the voting system cannot report a cast-vote record linked to each cast card. It obviates the need for re-scanning; it is simpler and more efficient than ‘hybrid’ audits; and it is far more efficient than batch-level comparison audits. There may be room to improve the efficiency of ONEAudit further by tuning the statistical tests it uses and by using stratified sampling. We show that tuning the tests by optimizing for the reported batch-level tallies or integrating over a distribution reduces expected workloads by 70–85% compared to the current ONEAudit implementation across a range of simulated elections. The improved tests reduce the expected workload to audit the 2024 Mayoral race in San Francisco, California, by half—from about 200 cards to about 100 cards. In contrast, stratified sampling does not help: it increases workloads by about 25% on average.