Funny when we were his age back in the day we really never wanted an OLD car no matter how redone it was, it still was an old car.
You can probably thank the 70's and 80's for that. I think it took a few decades of the cars getting crappy to make people appreciate what had come before.
I think to a degree, the '57 Chevy became popular pretty quickly as a used car. That same year, the Ford and Plymouth got longer, lower, and more impressive looking, while the Chevy was really just a facelift of a car that was 3 model years old...ancient history in those days. As a result, Ford outsold Chevy and Plymouth did well enough to knock Buick out of 3rd place, and build a record number of cars...a record that would stand until the 70's I think. For 1958, Chevy responded with a much bigger, lower, more impressive looking car. But suddenly, I think these bigger cars began to be associated with what your parents bought, and a lot of younger buyers just didn't want something that big.
It probably also helped that the '57 Ford and Plymouth were serious rusters, and I think build quality went down on the '58 Chevy as well, so more '57 Chevies simply survived. Anyway, there was enough clamoring for a '57-Chevy-sized car again, that for 1964, Chevy released the Chevelle, which was about the size of a '57.
Back in my college days, I remember one of the managers at the Denny's I worked at talking about some of his old cars. He had a '57 DeSoto Fireflite 4-door hardtop, that he paid $500 for in 1965. He said it was pink and white, although I'm guessing it was really more of a salmon. It was considered a "loser" car, because they didn't make DeSotos anymore for one thing, but I also suspect that nobody associated sky-high tailfins with "cool" in 1965. My Mom had a '57 Plymouth as her first car that year, and she hated it. That DeSoto would easily embarrass many cooler cars when it came to drag racing, but it still just wasn't cool. My manager sold it and bought a '57 Chevy convertible, also in pink, for $500. I dunno what his fetish for pink was! Anyway, it was a dog with its little 283-2bbl compared to that DeSoto's 341 Hemi, but it was a much "cooler" car.
I think it was the late 70's when '55-57 Chevies really started getting popular and prices started to take off. Probably thanks to movies like "American Graffiti" and such, and the big 50's nostalgia kick that started in the 70's.