The phrase "more periodically" is used to describe something that is done with increasing frequency. Some synonyms for this phrase include "more frequently," "more often," "at shorter intervals," "regardless," "with greater regularity," and "more consistently." Each of these expressions conveys the idea that something is happening more often than before. To provide more specificity, other options include "with increasing frequency," "on a more regular basis," "with greater frequency," or "at shorter and shorter intervals." These variations all help to clarify the meaning of the original phrase and can add depth and precision to your writing or speech.