It's decided that the Hawks in the JLA are the Hawks from the JSA, just older, wiser and helping out the new generation. Eventually, Hawkworld as an ongoing series bombs and they make Katar as yet another Hawkman. He was canceled soon after, and there hasn't been a Hawkman since. Grant Morrison wanted the angel Zauriel in the current JLA to name himself Hawkman in tribute, but editorial decided that the name was so tainted from multiple relaunches that they shouldn't use it. There currently isn't one.
Plans are afoot for a new Hawkman.
Let's hope they just start from scratch and give us a cool one. The JSA was the first super-team in comics. As the years went on, their WWII-era adventures were consigned to Earth 2. When the timelines merged in Crisis, it was stated that the JSA was active throughout the '40s and disbanded in the early '50s. Eventually, they were freed and returned to Earth.