"As mad as a mad hatter!"
In the 18th century, they use to use mercury to stiffen felt hats, and then would get mercury poisoning, which led people to go 'mad'. The 'mad hatter' character in 'Alice in wonderland" was inspired by a guy named Carter who knew milliners who went mad or had dementia.
"Throw your hat into the ring"
Make or take up a challenge, or demonstrate one's willingness to join an enterprise.
The ring in question here is a boxing ring. These, of course used to be circular spaces in a crowd of onlookers, rather than the square, roped 'rings' of contemporary pugilism. Any Jack the lad who fancied his chances in a bout would throw in his hat - presumably this was a more reliable way of putting oneself forward than just shouting over the hubbub of the crowd.
I like the one I found and use in my website - it's called 'Where did you get that hat?' and I think it's a perfect example of old fashioned wording, and it says what a lot of people think even today, when they see a hat they really like.
Not too sure where the word ' tile' come from in relation to a hat, but it does convay the meaning well - after all, a tile sits on top, just like a hat
This is not a saying, more a little rhyme
"She wears many hats." I am not sure of the origins but it simply means that a person can do many things or plays many roles in life.
Mad as a hatter...... Started centuries ago due to the hallucinogenic properties of the hat stiffener!