It is actually not the next docking bay, but the next after that.
Strangely enough, as you move out of the bay, a woman rushes past you with a large satchel and a blaster rifle. "Pardon me, 'scuse me," she says as she moves past. She seems to be heading toward the commotion that Kyle has caused.
As you approach Snith's bay, there is the sound of an explosion behind you, but it passes from sight what is going on as soon as you step into Snith's bay.
The man is standing beside his ship, apparently ready to make his exit. He has discarded his heavy trench coat and ragged outfit for a state-of-the-art flight suit. He turns at the sound of your voice as he pulls on his piloting gloves, his expression grim.
"So," Snith says. "You didn't die, then."