The point when a used car became an antique is a little nebulous, especially when you are talking about the origins of the antique car hobby. I really don't know much about the postwar hot rod and custom culture and the practice of modifiying antique cars, but I can definitely speak to the prewar origins of the collector car hobby.
The Antique Automobile Club of America (the first of it's kind) was founded in 1935, but the origins of the hobby go back a while before that. Before that, there were individuals - many now lost to history - who saw what were at the time just used cars, many obsolete a few years after they rolled out of the factory, and decided for one reason or another that they should be saved. Maybe it was the miserly, penny-pinching New England post office manager who purchased a used 1912 Packard in 1914 and refused to let it go for thirty years; maybe it was a sentimental scrapyard official who saved a magnificent aluminum-bodied Pierce-Arrow from the scrap drives of the World Wars; maybe it was a wealthy couple like Larz Anderson and his wife, who purchased a new car every year for thirty years, holding on to nearly all of them; maybe it was a car dealer who wanted to exhibit the progress his company had made from its' humble origins; maybe it was just a farmer who parked his car in a barn and forgot about it.
By the late twenties, people had begun to realize the importance of saving those machines from the formative years of the industry. In 1927, the Andersons began exhibiting their collection to the general public on their estate in Brookline, MA. By the thirties, several collectors had begun tracking down and buying up these early cars, determined to save some "orphans" - cars made by defunct manufacturers. One of these such fellows has been the subject of a long running feature in my own club's publication, the Horseless Carriage Gazette, entitled "How the Hobby Started." Art Twohy was a prominent lumber dealer in Los Angeles who began collecting pioneer-era cars in the early thirties, and used these curiosities to drum up business. He also lent them out for Hollywood features, parades, and the like. Twohy "had always been interested in automobiles, particularly old cars, and when I learned that really old cars were so scarce that they were being virtually considered antiques, I decided to start my own collection."
By the mid-thirties, the first groups dedicated to collecting and preserving these cars had emerged, starting with the AACA in '35. By the end of the decade, the club had 115 members, and in 1937 it began publishing its' club magazine.
The piece on Art Twohy is from the Horseless Carriage Gazette Vol. 76 No. 6 (which may be hard to get a hold of - If you'd like a scan, let me know). The Automobile and American Life by John Heitmann is also a great source on the topic. As always, don't hesitate to hit me up with follow-ups!