And on cards from the '50s its next to impossible, since there were no action shots, only posed ones. But notice I said, "NEXT to impossible."
I found a 1957 card that I can pin down to a three day span.
First, I know for a fact that the picture on this 1957 card was taken in 1957. That was the only season that the Cubs wore this style of road uniform. Since the picture for a '57 card was taken in the same year the card was released, I have to assume that it is from the early part of the season.
Second, I know that the picture was taken in Philadephia at Connie Mack Stadium. Behind Singleton is the famous "Spite Fence" that Connie Mack built to block the view of the ballpark from the houses across the street.
So it was just a matter of checking the schedule to see when the Cubs were in Philadelphia in the early part of the season. The answer? They had a three game set against the Phillies from May 2-4.
Another clincher for those dates is that Singleton only pitched in games in April, May, and one in September.
So a combination of '57 uniforms, Connie Mack Stadium, and games in April-May give me pretty conclusive evidence that the picture was taken on either May 2, 3, or 4, 1957.
And this one too!