The term you're looking for is Silver Age for that period. If you want to watch even older films, we'd end in the Golden Age of Cinema (a fine age... there's such a charm in the GA films although Silver and Bronze Age have produced fine films too).
Anyway, I enjoy many "old" films. There are a lot of enjoyable classics out there like Waterloo Bridge, Gone with the Wind, Nosferatu, Anatomy of a Murder, Red River, The Big Country, East of Eden, Shane, On the Waterfront, Chaplin's and Keaton's films among other things. Lots of quality stuff to choose from.
I personally think many of the actors in classic films could find work in today's Hollywood. Just look at George Sanders in Man Hunt (1940s version), for instance; he'd be right at home in today's thrillers with his charisma and screen presence.
I was mildly disappointed when I saw it; I guess it was because I read the novel which was superior (the film had to drop a lot of interesting subplots to only have the essential stuff). It's a fine film as it is, however, and Elmer Bernstein's music really captures the innocence, darker undertnes and inherent bittersweet quality of the story.
If you're a fan, check out the Caine Mutiny. It's a very interesting, thought-provoking film for Bogart with an enjoyable plot.