One of the many things Clint keeps secret is the fact that he is actually a mutant, not a baseline human. Surprisingly, his mutation has nothing to do with archery or his eyesight or his reflexes or anything like that.
Clint has a very unique and mild form of clairvoyance. Sometimes, on missions or in real life, he will get ‘flashes’ where he can ‘find’ people he concentrates on or just small indicators of how to best approach and interact with people, no matter their mood or temperament.
That’s why he and Natasha are so close; he just knows when she needs to let out some internal stuff or when she is unapproachable.