The phrase "out through idle curiosity" implies a sense of aimlessness or purposelessness in one's actions. When trying to find a synonym for this phrase, one might consider words like "mindlessly," "pointlessly," or "curiosity-driven." Other possibilities might include "unintentionally," "accidentally," or "haphazardly." Each of these words captures a different aspect of the original phrase's meaning, such as the lack of direction or the lack of intentionality. By choosing the right synonym, one can more accurately convey the intended tone and nuance of the phrase.