I agree that HAWX3 better stay in near future. It's connected to Splinter Cell and Ghost Recon very closely, after all.
As for the storyline, I can offer something like this:
Some years after HAWX2 US, Russia and EU are all in one big conflict beacuse of three-sided millitary conflict in some small coutry (McGuffinania, ha). Each side supported a differen party then conlict escalated and the big players got themselves into war.
You play as mercenary pilot (or member of McGuffinanian Air Force) who just happened to be in the middle of the conflict and must choose sides.
You side with one of the parties and fly some missions for them, then you (and your wingmen) notice that whenever one of the parties manages to get some advantage, something happens and the balance of power is restored and no side can win.
Then some time later Agent Drachev (Wolfhound), (alternatively - Sam Fisher) contacts you and reveals that the real cause behind this conflict are the bad guys who were manipulating General Morgunov and others into starting the conflict some years ago. (Probably, "Maggito", who were mentioned in SC: Conviction, and hinted at in HAWX2)
They are using the conflict for the distraction and while world's attention is brought to war, that can grow into WWIII, they're quietly taking control of all world's major finansical currents. Then they plan to reveal themselves as "peacemakers", who'll "resolve" the conflict, and gain not only financial, but also political support and becoming one of the most influental group in the world. (Taking over is a cliche, but getting as much power as possible is more likely and down-to-earth scenario).
So for next missions you take out the conspiracy's agents within all of three parties' forces. After that you (and mabye some allies from all three armies) go around the world and take out the conspiracy's members, who've been identified by Drachev.
Something like that. (I know, it's a bit Ace Combat-ish, but AC never failed us with stroy before).
Or to go into complete (and crazy) Sci-Fi and make the antagonists a group of powerful individuals, who believes, that humans can never understand each other, because they have two sexes and plan to create a mutagen the turn all humankind into hermaphrodites, while using the constant small conflicts around the world to divert attention form themselves.
You play as member of quick responce peacekeeping air force who is thrown in coflict after conflict, then somehow become aware of conspiracy's evil plan and then you're trying to foil it.
(Plain crazy, I know. Don't take this seriously. It's just came up, when I remembere the idea of "sharks with lasers on their heads", that was a joke wish fo HAWX2).