Despite attempts over the years by DC to simplify things by reducing the number of multiverses that they have to contend with, there has never been just one DC universe.
What about the Golden Age? You ask.
Remember Doll-man and Phantom Lady they were around back in the Golden Age and their Earth (Earth-X) was in a separate universe – that is “X” as in Quality Comics. See Justice League vol. 1 #107 (Oct-1973)
Currently, there is not a Hawkgirl existing in the main continuity of the “Rebirth” universe, but there is one existing way off in a corner of the multiverses in a separate continuity. She lives in the universe of the DC Super Hero Girls line of comic books. Recently introduced in the past few months, this series has been “developed for girls aged 6-12 and it is made up of a diverse line-up of female characters as depicted as relatable teens”.
As you might guess there is a clique of girls and Hawkgirl is not one of them. The main characters are Wonder Woman, Supergirl, Batgirl, Harley Quinn, Bumble Bee, Poison Ivy, and Katana.
But that is not to say that Hawkgirl is not in there. She just does not always hang-out with Wonder Woman and her BFFs. See if you can spot her on the cover of this $10 comic book.
As you can see below, the Hawkgirl character appears to be quite ‘fetch’ with designer (literally one of a kind) sunglasses perched on her head. From what I gather she is supposed to be Latina because she occasionally uses Spanish words and phrases.
If she is Latina, there is some continuity there. As recently as 2016, there was a Hawkgirl living in yet another DC universe. Her name was Kendra Munoz-Saunders. She was in the Earth 2 series and was a member of the JSA-like group named “The Wonders of the World “.