When we finally do get a look at his mug, he's not wearing a hat. His tumbler of whiskey is in the frame, but his head isn't. He crosses the room out of focus, moves past the camera, and when we see a reverse angle, he's standing behind and to the side of Leo. Tom is the one who put the cubes into the glass and poured himself some whiskey. We don't see Tom, our main character until the next shot, where he appears behind the bald head of a man (Johnny Casper, played by Jon Polito) who's delivering a lecture into the camera - or just past it - about friendship, character, ethics.
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.)
