He was selected to be absent from Bartlet's second State of the Union address as the Designated Survivor, the government official who would succeed to the Presidency should a catastrophic event hit the address. He gave the President a gift; the Constitution translated into Latin (the product of a school project by students).
President Bartlet gave him a summary of his own view of the Presidency, overheard by Leo McGarry, including the idea that a best friend who is smarter than oneself is the ideal Chief of Staff.
Tribbey reappears briefly, still as Secretary of Agriculture, in "Twenty-Five", when he is one of the members of the Cabinet giving their 'consent' to the President's decision to relinquish his position temporarily. He is even referred to "7A WF 83429" when President Walken declares that he himself is but one prime rib dinner away from a heart attack and "I don't want the Secretary of Agriculture bein' named as the next President of the United States", so the White House senior staff need to get their act together on choosing a VP.