The movie is set into motion with a close-up of three ice cubes plopped into a glass tumbler. The other one is on the head of his boss and friend, Leo O'Bannon (Albert Finney). The hat in all three close-ups, hat belongs to Tom Reagan (Gabriel Byrne). Take a look at the four shots from Joel and Ethan Coen's 'Miller's Crossing' on this page: three close-ups of the same hat and a long shot of another one with a body under. Warning: This post (and the short film montage/hommage I put together to accompany it, above) may contain spoilers.
(My final contribution to the Close-Up Blog-a-thon at the House Next Door, which just wrapped.)