I am not sure there is any such expression in English. The expression is "to throw one's hat in the ring." This is an idiom that comes from the sport of boxing and goes back to the mid-1800s, when a person who wanted to challenge the champion would throw a hat into the boxing ring to signify the desire for a match. It became a political idiom, referring to a new candidate letting the public know that he (and later, she) was going to start a campaign for office and try to defeat the incumbent. The inauguration, however, is when the person who was elected is officially sworn in. Thus, your expression would not make sense, because throwing your hat in the ring means you have not yet been elected and are offering a challenge to the incumbent.
The next Inauguration Day will be January 20, 2009. hat is the day when the new president is sworn in.
The one who said toss the hat into the ring.
John F. Kennedy.
Mr. Song Millinery - Detroit, Michigan
John F. Kennedy in 1961. He did not wear it during the swearing-in, but he wore it to the event.
Wore what? shoes? socks? a hat? a scarf? a pinkie ring? a nose ring? his hat backwards?
PINK
It comes from the boxing ring In the early 19th century, anyone who wanted to challenge a boxer would throw his hat into the ring
Andy's Hat in the Ring - 1924 was released on: USA: 6 October 1924
Hats were worn for dress occasions for many years and president wore them . Lincoln was noted for his tall stove-pipe hat. Eisenhower wore a Homberg to his inauguration instead of the more traditional top hat, but Kennedy's inauguration, he and Eisenhower both wore the silk tophats. Reagan wore a cowboy hat on his ranch.
Theodore Roosevelt
PINK