When we're looking for an antonym for the phrase "most near-at-hand," it's important to focus on the concept of distance or proximity. Some possible options might include "farthest away," "distant," "remote," or "removed." These words all suggest a greater amount of distance or separation between oneself and the object in question, and can be used to describe things like locations, people, or objects that are further away or more difficult to access than others. Ultimately, the antonym for "most near-at-hand" will depend on the specific context and the type of distance or separation that you're trying to describe.