Unlike today, the military didn’t announce whether they had changed their Defense Condition after Kennedy was killed from DefCon One to Two to Three to Four to DefCon Five which means war. (I’ve never read anywhere that they did but I’ve hardly read every book on the subject.) Based on public information available in the 9/11 Commission report, the United States military has never gone beyond DefCon Three and we have only gone to DefCon three on three separate occasions, one of them being 9/11. But I would wager a dollar to a doughnut we went to a higher state such as DefCon Four (nuclear or major war imminent) when Kennedy was shot.
I don’t know when it ended but in those days and through the 1970s and 80s, there was always a large airliner, specially built for the Air Force, which had a one or two star general aboard and a lot of communications people. In the event of the bomb being dropped, the President could issue orders through this communications unit to American forces in the event other means of communication had been knocked out. There were several of these planes, of course, and the one aloft could not land until the relieving aircraft took off and achieved its directed altitude and course.
The internet, by the way, was invented by DARPA (Defense Advanced Research Projects Agency) as a way for critical parts of the government to communicate in the event of an air burst of one or more nuclear missiles which would fuse the electro-magnetic equipment used by existing telephone networks. No one had any idea what it would grow into. After the Feds set it up, universities doing research for the defense department began to use it and the rest is history.