The term "puts a tail on" typically refers to the act of following someone surreptitiously. Antonyms for this term could include "avoids," "ignores," "dodges," or "evades." If one is looking for the exact opposite of "puts a tail on," then they might say "gives someone space" or "leaves someone alone." Other possible antonyms could include "trusts," "respects," "disregards," "neglects," or "forgets about." Essentially, any word or phrase that indicates a lack of attention or pursuit could serve as an antonym for "puts a tail on.