A hair's breadth is a very small and precise measurement, usually referring to a very thin width or distance. Some synonyms for hair's breadth could include a whisker, a micro-inch, or a razor's edge. Other terms that could be used as synonyms for this phrase could include a fine line, a sliver, or a nano-margin. Each of these terms conveys the same sense of preciseness and thinness as the original phrase, allowing for a variety of ways to express this concept in written or spoken language.