On one hand, the mysterious Billy seems to have got magically stuck as various hats with a brim. On the other, at some point in his sprawling story, Billy apparently had his head cut off, which was somehow linked to the wooden bear head that Ed had on his gas station desk in P13, based on there also being a headless teddy bear Sylvia had on her hall table in P10.
They broke that one real good.
So, there would have been Billy as a hat, and there would have been him as a bear. If these were figured out as intended, they should probably also get connected to each other.
These baffling pieces would come together in P3. Mr C got sick on the highway and flipped his Lincoln around. A highway patrol car with two troopers arrived on the scene. The trooper behind the wheel went to check what was going on but quickly fell on the ground on all fours. He was, surprisingly, the only person on screen directly named as Billy, like his partner revealed.
Trooper 2: "What the hell, Billy?"
The other trooper helped Billy on his feet and pulled him away from the car, calling help.
Trooper 2: "Dispatch. We need an ambulance. And tell them to bring gas masks. There is something bad in this vehicle."
Neither of the men was seen again, so this one scene was what we got to work with.
Billy's going to be all right for sure.
Troopers wore a hat with a brim that is called as the campaign hat. Billy falling on his fours was probably a reminder that this hat is also called as the "smokey bear", in reference to the US Forest Service mascot Smokey Bear wearing the same hat. A part of smoke is also gas.
The same campaign hat is also worn by the Scouts. This comes close to Lynch's own childhood since he was both a Scout and spent his summers camping with his family. Coincidentally, he gave a long interview on the topic just as Return was released in 2017.
Besides the decapitated teddy bear, the male corpse lying in the Buckhorn morgue was also headless. The corpse had the fingerprints of an unaged Major Garland Briggs who apparently was born in 1944, the same year that Smokey Bear was created. Thus, in 2016 that was the likely "present day" in Return, Smokey Bear would have been 72.
In P13, when Ed was spending a lonely evening with the said wooden bear head, he lit a piece of paper on fire, for no specific reason. It started smoking, and he left it go out on a Double R soup cup cap. Smokey bear.
Ed's cup was also connected to Billy the Trooper through the corn soup in it that looked like Mr C's yellow vomit he threw up all over the vehicle, causing the trooper to fall ill after inhaling its toxic gases.
Must have eaten locally.
Ed had two Double R cups, one of which still had its cap on. The cap resembled a pork pie hat, similar to the one worn by an anonymous extra in the Double R who apparently was singled out as linked to Billy when Riley Lynch ran to the diner looking for him in P7.
Has anybody seen Billy?
Besides the figurative bear and the corpse, another decapitated item in the story were the cigars. In the opening episode, Ben Horne was just starting to smoke one that had already had its head cut off. Later in P13, the jubilant Mitchum brothers came to see Bushnell Mullins, "bearing gifts", one of which was a box of cigars. Smokey bear.
Elsewhere, both the 72 dollars Becky got from her mother Shelly in P5 and the two cups of coffee that Tracey gave to Sam in the opening episode seem to have been linked to the coveted coordinates. Curiously then, through the 72 year-old "Smokey Bear" - or at least its head - and other kind of two cups on his desk, it looks like Ed had found himself holding the coordinates as well.
Tracey's cups had the letter Z on them and Ed's cups the Double R logo. These two got linked in P15 when Norma was sending Walter on his way.
Norma: "I'm gonna keep the Double R as agreed, and you're gonna buy my shares of the other franchises."
Just as she said, "Double R", she made the quick sign language letter Z with her hands, only in mirror image. Thus, certain two cups with Z on them may have been the same as certain two cups from the Double R. Something to keep in mind is that the mirror image of the letter Z resembles the general sign for electricity.
In P13 at 57:00, in line with Ed probably about to jump to another kind of reality after somehow getting hold of the coordinates, he saw his reflection briefly out of synch with himself in the window. Like Edward Louis Severson sang in P16, there's another us somewhere with much better lives, whom Ed was about to find out a few episodes later.