I'm going to point to the It Can't Happen Here model, despite a voice telling me that it'll be a bad idea. America democratically elects a dictator (Take your pick, the 20's and 30's were littered with them) in a worse depression, but things go bad and he goes mad with power, and builds a massive army to manifest destiny the rest of the Americas. The Army overthrow him after a year or so, realizing how insane he is, setting up a military Junta out of Washington to reform the American Government (Let's say John J. Pershing). Pershing keeps the Junta running until his death in '48 (or earlier), during which he slowly phases Civilians back into the Government. Upon his death, a new election is held. I know it's not exactly what you wanted, but it does give a reason for the Coup, and it's the best I got.