Obviously, any continuation of London's "To Build a Fire" is pure speculation, and you can extend the story in any direction you want without risking error.
I can give you a couple of directions you might want to take the story. A continuation of the story could center on the dog. London liked to write stories from the dog's perspective, so this would even fit London's practices. Maybe getting to the next warm spot is easy for the dog, but once its there, trouble develops.
Or, if you choose, you could probably pick up the story with the discovery of the body. You could then center on whoever discovers it.
Again, you can write whatever you want. These are just a couple of ideas.