The word "most everpresent" refers to something or someone that is always present or found everywhere. The antonyms for this phrase would be words or phrases that describe the absence or rarity of something. Some possible antonyms include "scarce," "infrequent," "rare," "uncommon," "sparse," "occasional," "intermittent," "occasional," "sporadic," and "scattered." These words describe things or people that are not always present or easy to find. They are the opposite of "most everpresent," which implies a constant, ubiquitous presence. Therefore, the antonyms of "most everpresent" signify a rarity or a lack of continuity.