How does Hemingway use the technique of telling the whole story without looking at the big picture to reveal how veterans feel about the war?

In The Sun Also Rises, Hemingway has Jake Barnes narrate a story in which people watch bullfights, travel through the South of France and Spain, and indulge themselves in ample amounts of food and drink. Superficially, none of this seems to have anything to do with the First World War and all of this activity is quite mundane. However, the key to understanding Hemingway's method of storytelling is to pay careful attention to what he does not narrate and what he does not allow his characters to say.
This latent narrative is most apparent, I think, in the final pages of the novel when Jake and Brett Ashley, the love interest with whom he can never consummate a relationship, due to his war injury, have lunch at Botin's—"one of the best restaurants in the world." Though Brett does not eat much and never does (our modern sensibility would suggest that Brett starves herself to fit the era's fashion for slimness), Jake eats "a very big meal and [drinks] three bottles of rioja alta." This insistence on enjoying life through sensory pleasures is one of Jake's positive responses to his war experience. On the other hand, his incessant drinking, and that of the other characters, is an attempt to numb the pain from the experience.
After the meal, Jake orders two more bottles of the wine and Brett implores him not to get drunk:

"Don't get drunk, Jake," she said. "You don't have to."
"How do you know?"
"Don't," she said. "You'll be all right."
"I'm not getting drunk," I said. "I'm just drinking a little wine. I like to drink wine."
"Don't get drunk," she said. "Jake, don't get drunk."

The implication is that Jake drinks to forget about his injury, which prevents him from sleeping with Brett, who has just broken up with a bullfighter (who could not satisfy her sexually but who blamed her for his inability to perform sexually):

"You know," Brett said, "he'd only been with two women before. He never cared about anything but bull-fighting."
"He's got plenty of time."
"I don't know. He thinks it was me. Not the show in general."

To get more specifically at your question, Hemingway shows us, through the lens of the relationship between Brett and Jake, how the war permanently impacted people's lives, making it difficult for them to have the relationships that they wanted. This unfulfilled desire kept them in permanent pursuit of a lust that they could never quite satisfy. They consume as much as they can (and this behavior may also explain the consumerist boom in the 1920s) to forget about the needs that they cannot satisfy.
In the end, Jake and Brett get in a taxi together and ride through Madrid, so that they can see the city. They sit close to each other and Brett says that they "could have had such a damned good time together," to which Jake responds, "Isn't it pretty to think so?" The romance they wish to have with each other can only exist in their dreams; the war destroyed their future before it could even begin.