The factor of four increase in the LHC luminosity, from 0.5x10(34)cm(-2)s(-1) to 2.0x10(34)cm(-2)s(-1), and the corresponding increase in pile-up collisions during the 2015-2018 data-taking period, presented a challenge for the ATLAS trigger, particularly for those algorithms that select events with missing transverse momentum. The output data rate at fixed threshold typically increases exponentially with the number of pile-up collisions, so the legacy algorithms from previous LHC data-taking periods had to be tuned and new approaches developed to maintain the high trigger efficiency achieved in earlier operations. A study of the trigger performance and comparisons with simulations show that these changes resulted in event selection efficiencies of >98% for this period, meeting and in some cases exceeding the performance of similar triggers in earlier run periods, while at the same time keeping the necessary bandwidth within acceptable limits.