The short answer to your question is: YES, they did. Herodotus and Josephus both reference and footnote many more ancient historians and their work. Rulers throughout the eastern Mediterranean are among them.
Unfortunately, much of it has been lost to history in such events as conquest/destruction among nations and tragedies such as the fire that destroyed the Great Library of Alexandria.
Added 8/5/21… Since a few ppl have asked that I actually name some of those earlier historians, here you go: Hecataeus of Abdera, Ctesias of Cnidus, Ephorus, Theopompus, Hieronymus of Cardia, Duris of Samos, Diyllus, Philistus, Timaeus, Polybius, and Posidonius. (Listed roughly from older to newer.)
(After a recent downsizing move, most of my books remain in storage so the preceding list is from Diodorus, not Herodotus.)