After appearing during World War I just long enough for plot development, Cole emerges in 1996. He kidnaps Railly and the two of them take off for Philadelphia, the origin of the virus and home of the Army of the 12 Monkeys. Cole locates them, and their leader, Goines, before disappearing to 2035 again. Note it is an ascent, not a descent, though. He's a good observer and strong mentally.
Televisions figure prominently in the film.
Contrast Goines's story about germs and Cole's eating of the spider. It's too much for even Goines to take. Compare Goines's facial reaction to that of Madame Scientist when Cole tells her about eating the spider. During Cole's escape attempt, the security guard tells him to use the other elevator. Note that we never actually get to witness his arrival or departure.