What happens in the middle part of the story to make the men believe they will soon be rescued? How do the men react to this?
At the end of Section II, the men spot the lighthouse on Mosquito Inlet. In Section III, they continue to make progress, albeit very slowly. The captain mentions that he can see a house on the shore and this is a "house of refuge." The captain says that the lighthouse keeper should be able to see them by this point. But the oiler is skeptical that someone on land would even know to be looking for their small boat, given the improbability that word of their shipwreck has reached land. They continue to wait to be spotted by someone on shore, but to no avail.
Finally, near the end of section IV, the men see someone on land. This is the moment, about half-way through the story, that the men have some hope of rescue. The man on shore waves to them and more people appear on land. They think that it will only be a matter of time before a boat is sent out to rescue them.
"Ah, now, we're all right! Now we're all right! There'll be a boat out here for us in half an hour."
However, the men can not understand the gestures of the people on land. The men react with hope and then frustration, alternating emotions of optimism and despair. They think they see a lifeboat but then it appears to be an omnibus. These alternating reactions continue until dusk obscures anyone on land that they might be able to see.