by a hair's breadth; barely hanging on; still having a tiny chance
Idiomatic expression meaning that one is barely surviving or has only a very slim chance left, often used with verbs like 残る or つながる.
I passed the exam by a hair's breadth.
The company barely avoided bankruptcy by a thread.
危機一髪 emphasizes a critical moment of danger, while 首の皮一枚 focuses on barely surviving or having a slim chance left.
間一髪 is used for a very close call in timing, whereas 首の皮一枚 is more about a precarious state or slim hope.
Literally 'one layer of skin on the neck,' evoking the image of a neck barely attached by a thin piece of skin, i.e., a very precarious situation.