Been thinking about this lately since the Black Adam movie. A JSA movie seems an obvious spin-off, but I feel a solo Hawkman movie, starring Aldis Hodge, is a solid option as well!
I'm skeptical of race-bending in general, but Hawkman is a character where it kinda works, and Hodge's performance, while stopping short of mind-blowing, was pretty darn good. But it also made me very curious about this take on Carter Hall and his backstory.
So here's how I'd do a Hawkman movie...titled Hawkman of course
The film would be primarily set in 2017, sometime after the events of Justice League. Carter Hall is an archeologist and a consultant to ARGUS. His speciality is in tracking and investigating alien artifacts. Carter is recruited by Waller for a covert discovery mission in Egypt, to unearth what could potentially be an alien spacecraft that crash-landed on earth in ancient times. The craft is discovered in the tomb of Prince Khufu. Within it, Carter finds a device called the Absorbascon which floods his mind with memories of an alien world of winged people and of past lifetimes in Ancient Egypt and throughout world history.
As Carter deals with these strange visions and memories, some of the alien tech recovered by ARGUS is stolen by a mysterious organization, led by Carl Sands AKA the Shadow-Thief. Carter uses the alien armor and wings he'd taken from the spacecraft, powered by the Nth metal, to attempt to stop the theft but fails. He works together with a young ARGUS agent named Kendra Saunders to attempt to discover who's the mastermind behind Sands and what they intend to do with the stolen tech.
Carter's appearance in armor also leads the media to speculate about an obscure long-forgotten hero named Hawkman, who was a member of the Justice Society. Carter starts to investigate Hawkman, and has visions of being him and fighting alongside the JSA in the 70's and 80's. At some point, he encounters one Kent Nelson who greets him as an old friend. It's at this point that Carter learns the truth from Kent.
Carter is the reincarnation of Prince Khufu, who along with his lover Chay-Ara, discovered the crashed ship from the planet Thanagar. Hath-Seth intended to use the alien tech to conquer Egypt and dethrone Khufu. However, during their battle, in which all three perish, they are exposed to the Absorbascon, which potentially lead them into a cycle of rebirth across centuries - usually with the same bodies, but sometimes with different ones. Kent takes Carter to the latter's estate, which he entrusted to Kent and the rest of the JSA, to pass on to his next incarnation. Carter finds there the prototype Hawk-cruiser, which he is able to fix and get working using his newfound understanding of Thanagarian tech.
In 1986, Hawkman battled Hath-Seth's reincarnation, Dr. Anton Hastor, but both killed each other in battle. We learn that in the present-day, Hastor was reincarnated as woman named Helene Astor, whose memories were re-awakened the moment Carter, in his latest incarnation, discovered the spacecraft and activated the Absorbascon. Helene swiftly reunited with Hastor's cult and put into motion the plans to steal the Thanagarian tech (through their agent Shadow-Thief, who'd been given some Thanagarian tech Hastor had acquired back in the 80's) and harness it to fulfill Hath-Seth's original goal of world domination.
Carter, now having embraced his identity as Hawkman goes into battle with Helene Astor, along with Kendra, who reveals herself to be a deep-cover Thanagarian agent named Shayera Hol (whom Carter vaguely believed might by Chay-Ara reborn). The Hawks battle Helene and manage to defeat her, destroying her consciousness seemingly permanently by harnessing the Absorbascon. In the aftermath, Shayera takes the Absorbascon back with her to Thanagar, with the possibility that she might one day return, while Carter goes back to working with ARGUS, but this time as a superhero and field operative, and he and Kent (and Al) get together to talk about old times and the possibility of reforming the Justice Society someday...