Not having read the story, we can only hypothesize. The moral could be about risk - taking and its consequences. If in the story someone gives up the 'bird in the hand' (a sure thing) in pursuit of more but ends up with nothing, it could be teaching us to be cautious about sacrificing what we already have for the sake of something that might not come to fruition. It could also be about the wisdom of recognizing the true value of things we already possess rather than always chasing after the unknown.
As I haven't read the 'a bird in the hand short story', I can only make a guess. Maybe the moral is related to being content with what you have. If the story involves a character who has a secure option (the bird in the hand) but is tempted by other seemingly better options (birds in the bush) and then realizes the value of the initial option, that would be a common moral in such stories.
Again, without knowing the story, it's difficult to determine the exact moral. But if it follows the common idiom 'a bird in the hand is worth two in the bush', the moral might be about the value of certainty and not being too greedy for more when you already have something good.