After COIE, they were stuck with characters who were A) essentially the same character in the Golden and Silver Ages (Batman, Superman, Wonder Woman), B) only from the Golden Age, C) only from the Silver Age, D) in both the Golden and Silver Ages, but were different characters.
The ones from B/C were largely fine to keep going forward. The A group usually got new backstories, where it was either streamlined (Batman) or basically brand new (Superman, Wonder Woman).
The D people were the biggest problem, but they confounded it by not having a consistent approach. Jay Garrick's Flash had existed on the same Earth as Barry Allen's. Alan Scott's Green Lantern got his power ring from one that had been tampered with by a member of the Green Lantern Corps. And then there were the Hawks...
So, the Golden Age Hawks, Carter and Sheira Hall got shuffled off with the rest of the Justice Society to battle endless in the realm of the Norse gods to prevent Ragnarok (don't ask). The Silver Age alien versions, Katar and Shayera Hol disappeared completely. They were replaced by new versions, Katar Hol and his partner (not wife) Shayera Thal. That version was like something out of a Reagan Era action movie, where Katar was going around being a brutal policeman and using guns.
Eventually, they did this incredibly convoluted thing where it was revealed that all of the versions of the Hawks were reincarnated versions of each other, which actually kind of works. This lead to the resurrection of Carter Hall. Somehow. They eventually introduced a new Hawkgirl, Kendra Saunders, who had committed suicide, and in the process got possessed by Shiera Hall, who I think was supposed to be her aunt. This had the very creepy effect of mixing up Shiera's memories with Kendra's, and left Carter in love with her and trying to convince her to be in love with him.
Shiera actually got resurrected in the Blackest Night event, but I don't think they ever actually did anything with that, and to the best of my knowledge, the only version that has been seen since the New 52 and into Rebirth has been Kendra.
I have a headache now.