There is a song with title {hajimete no asa} and another song with title {hajime no hi}. There is also a manga title {hajimete no aku}, and not to forget the popular boxing manga {hajime no ippo}.
So what is the difference between 「はじめてのX」 {hajimete no X} and 「はじめのX」 {hajime no X}? How do we choose to use one over the other?