Well lets just say I can see the massive surplus of able bodied young men flooding the civilian sector and I know job prospects are grim since they sucked before I enlisted. How likely is it that I can stick around in the military?
I can't say for sure how likely it would be, but I know of anecdotal examples (e.g. some of the guys from Band of Brothers) of people who volunteered to stay in uniform after the war and seemed to have no trouble doing so. I came across a master's thesis that corroborates this, and also suggests that the overwhelming majority of soldiers wanted to leave the Army and get home quickly, and that the Army had some trouble filling its ranks in the immediate postwar years, even though it had shrunk from over 8 million men to just 600,000.
So, I'd say your hypothetical soldier would have had an easy time staying in if he wanted to.