The very angry teacher sent him home.
versus
The very-angry teacher sent him home.
. . .
Why might "very-angry" be unacceptable while "second-highest" might be preferred?
That's difficult to explain, but I will venture to say that the phrase "very angry" is different from the others because it is not a compound adjective. "Very" is an adverb, so that "very angry" is just an ordinary adjectival phrase.
The exception to this exception is the use of "well" and "ill" as components of adjectival phrases. They are adverbs too, but it is common to write "well-done steak" and "ill-gotten gains." According to the Chicago Manual of Style, as I recall, they belong to a special category because they are monosyllabic adverbs.