I'm not entirely sure if I understand your question, but the usage of "al-Yaman" to refer to an area in Southern Arabia is ancient, and might actually have been the word that was translated to produce the Greek and Roman term "Arabia Felix."

The narrowing of "Yemen" to refer to the contemporary state of Yemen is much more recent and involved various border disputes into the 20th century, including the Saudi conquest of Najran in the 1930s.

Source-wise, see the entry on the etymology and limits of Yemen in the entry for "al-Yaman" in The Encyclopaedia of Islam.