Tom's external conflict takes up most of the story. Essentially, the problem is that an important document is blown out the window by a draft and he has to climb out on a ledge eleven stories above the street in order to retrieve it. The document is "a creased yellow sheet covered with his own handwriting."
Of all the papers on his desk, why did it have to be this one in particular! On four long Saturday afternoons he had stood in supermarkets counting the people who passed certain displays, and the results were scribbled on that yellow sheet. From stacks of trade publications, gone over page by page in snatched half-hours at work and during evenings at home, he had copied facts, quotations, and figures onto that sheet....All were needed to support and lend authority to his idea for a new grocery-store display method, without them his idea was a mere opinion.
The author, Jack Finney, takes pains to show the importance of that one sheet of yellow paper. Finney also takes pains to describe Tom's reluctance to risk his life by climbing out on a narrow ledge and inch his way over to the place where his precious memo has gotten stopped by a projecting blank wall and held there by the breeze. Finney specifies that the paper is only five yards from Tom's window.
Finney devotes five full paragraphs to describiing how Tom finally talks himself into taking the risk of climbing out his window onto the ledge.
To simply go out and get his paper was an easy task--he could be back here with it in less than two minutes....The ledge, he saw, measuring it with his eye, was about as wide as the length of his shoe, and perfectly flat. And every fifth row of brick in the face of the building, he remembered. was indented half an inch, enough for the tips of his fingers, enough to maintain his balance easily.
The worst part of Tom's problem is the awful height. He knows it could be fatal for him to look down at the street because he could be overcome by vertigo and fall off backwards with a scream. He keeps his face pressed against the brick wall of the building as he shuffles along the ledge and finally reaches the yellow paper. But then he is forced to look down in order to get his hand on it. And at this point he is horrified by the insane situation he has placed himself in. The sight makes him lose his nerve. His legs turn rubbery.
He saw, in that instant, the Loew's theater sign, blocks ahead past Fiftieth Street; the miles of traffic signals, all green now; the lights of cars and street lamps, countless neon signs; and the moving black dots of people. And a violent instantaneous explosion of absolute terror roared through him.
A city like Manhattan can be a terrifying spectacle from many points of view. It can make an individual feel tiny, insignificant, lost, overwhelmed. The description of Manhattan from a ledge eleven stories above the street is the high point of the story. The mention of the fact that the miles of traffic signals were "all green now" is a nice touch. In a minute or less those traffic signals will all turn red and the change have a disorienting effect on the lone man clinging to the side of the building.
Tom knows he can't stay where he is at because his legs will give way and he will fall to his death. At the same time, he can't make it back to the window of his apartment because he is paralyzed with fear. He tries lighting matches to signal neighbors for help--but no one notices his plight in this cold, indifferent city.
The main external conflict in Contents of the Dead Man's Pocket, is that he is literally hanging from the edge of the window. He must figure out how to get off of the ledge literally. He is also suffering from internal conflict in that if he does not find some balance he is going to ruin his marriage. You read every thought that he has and while on the ledge realizes the changes he needs to make, and as he does this he also figures how to literally get off the ledge.