Most ill-founded means the most unsupported or baseless. Antonyms for this phrase would be words that convey the opposite meaning. Some suitable antonyms for most ill-founded include: well-researched, well-grounded, well-founded, well-supported, well-based, well-substantiated, firm, robust, solid, and sound. When making an argument, it is important to back up claims with sufficient evidence and reasoning. Otherwise, the argument may be viewed as most ill-founded. Therefore, using appropriate antonyms to assert the strength of one's arguments can help to persuade others that a position is not only reasonable but also logical and well-founded.