The Alley Awards were the comic equivalent of the Academy Awards during the 1960s although for some reason they died out after 1969. Using the information found here, the best comic covers of the 1960s were:
Also a very significant book, as we have discussed in the past.
There seems to have been a push by fandom to help out DC's resurrection of this character, as Hawkman was also chosen as best hero of 1962, despite the fact that sales of his tryout issues were insufficient to launch a solo title yet.
In 1963 there was no award for best cover per se, but this cover won for best single illustration:
I'm as baffled as you probably are by that one.
The New Look Batman picks up a win for Carmine Infantino.
An effort at reviving two Golden Age heroes. Although both this issue and Brave & Bold #62 were terrific, they did not sell enough to justify continuation.
Al Williamson's take on the comic strip classic. Various efforts were made to bring Flash Gordon to the comic books, but none ever succeeded. Most of the major publishers tried at least once--DC, Harvey, Marvel, Dell, Gold Key and King (which produced this attempt).
Neal Adams picks up the first of many awards. Note that this is the only cover chosen in the decade that has word balloons. Update: DOH! As pointed out by Snard in the comments, the Flash #123 cover also has word balloons.
Jim Steranko checks in with a memorable Nick Fury cover.
Steranko wins for the second consecutive year.