Hana is the bartender at The Freezer but she isn't the one that brings you back to the safe area. If you freeze in the tundra, before seeing Hana you should see a scene with whoever actually rescued you. If you're not seeing any of those scenes, then your copy of the game is bugged. I know there was a version in the last 2-3 months that broke the rescue scenes but it should be fixed now.
There are three rescuers that will bring you back to The Freezer. Jerome, Jerynn, and one other girl whose name I can't remember. If you get rescued by Jerynn repeatedly then the scenes escalate pretty wildly and her bad end is pretty great. Jerynn's sequence is my favorite part of the whole game.
Dreghan will sometimes rescue you and bring you to the Korg'ii hold.
I honestly don't remember ever seeing Vark rescue you though. Maybe there's some combination of stats and freeze location that you need but idk what that would be.