Military and prison adjective for describing someone nearing the end of their tour of duty, enlistment or incarceration. A short-timer is someone who has less then 90 days to serve. A Korean War veteran told me these terms were in wide use in the late 1940s; they have been in continuous wide use in the military since then.
Short was used long before there were compact cars. It goes back to at least the Forties on the Southside of Chicago. It means ANY car. It was from the vernacular of hotrodders and black gangsters. It probably derives from "hot short", which evolved to "hotwire" ... a car.