I cannot remember having seen anything written about it, which is why it has taken a while to find an answer to your question. And even now the answer is based on a bit of educated guesswork so don’t bet your shirt on it.
While discussing matters pertaining to deer with a friend in the Scottish borders, he said he had noticed this same phenomenon and in conversation with other experienced stalkers their impression is that the hair seems to be firmly attached when the deer is shot without being alarmed first, but the hair comes out in handfuls if the animal is alerted.
A chat in front of a log fire, whisky in hand, might have helped but over the phone the only idea we could come up with is that it might be a result of adrenaline triggering the flight reaction, which then loosens the hair to prevent a predator getting a firm grip. It seems possible but I don’t know the answer.
So when next you try to shift a deer whose hair is coming out, please give it some thought, consider the circumstances, and then let us know what you think.