So it's safe to say now that Betty Briggs is not Betty, Hastings' secretary.
Interesting info we got from Hastings, though.
So Major Briggs at the time of the season 2 finale was having visions of the future, enough that he was able to leave a secret message for Truman, Hawk and Bobby. And he knew that some sort of door would be opening at 2:53 on 10/1 and 10/2 of that year.
We know that at some point after meeting with Evil Coop 25 years ago, Major Briggs went into hiding in some sort of "alternate dimension" - most likely the White Lodge, since he had been there once before in season 2 and we see his disembodied head floating there in Part 3.
In present day, somehow Hastings and Davenport are able to access the place where Briggs is hiding, and he asks them for coordinates leading to a new place, fearing that he is no longer safe where he is. They find the coordinates he needs on a military server, and deliver them to him. Major Briggs starts floating (like the Giant did?) but before he can escape, "a lot of people" appear and that's when Davenport is killed (and presumably Briggs as well). "A lot of people" could be the group of Woodsmen, which would explain why they're still lingering around Hastings and Briggs' corpse.
So now Evil Coop wants the coordinates that Hastings gave to Briggs. These could be the same coordinates as "253 paces east of Jack Rabbit's Palace", since we know Briggs had visions of the future and might have been able to foresee his meeting with Hastings. Plus Lynch and Frost have helpfully labeled the Jack Rabbit's Palace note with the same symbol from Evil Coop's playing card - "This is what I want." And it would make narrative sense for Evil Coop's ultimate destination to be Twin Peaks, so that the climax of the story can occur there.
I'm just not sure where the Jack Rabbit's Palace "door" leads...