If I had to guess, especially based on the other comment including a line from an early script, it probably died on the editing room floor, was not fully thought out in the writing or they supposed you could presume it yourself based on other things occurring in the story. (he says he has the book and will get it to her, because he knows how much it means to her after picking it up at the train station) You see early on that he knows the new place she lives was still under construction. You also see that he is apparently able to change the future by planting a tree in front of it before it is finished, at which point the tree just appears over her as she's entering the building. They show this to the 'viewer' but it's not commented upon in their letters. This could be to add 'drama' in that neither of them would fully know that it was possible to change the future, including her at the end when sending the letter back. (it creates more suspense if it's an act of desperation for her) It's like how they don't actually show him 'reading' that last letter she sent but you do see the flag go down after we saw that - on that valentine's day when he allegedly was the one in the accident, he first went to the lake house to again look in the box. So obviously the flag going down means he did read that last letter, presumably before the accident would have occurred. 999
I'm frankly surprised they felt the need to show the scene of him hesitating on the curb, but they apparently thought it was the only way to tie together what had already happened with what was about to happen.
Personally, I think they could have found a more interesting way to explain some of the events than having him be the one that died in the accident. It kinda creates a paradox also in that he only knew of the date because she mentioned the accident. Thus he only remembered because of something he hadn't decided to do yet and only knew to do because she remembered it.
Cute story, but it could have been written better and probably should have included at a bare minimum an earlier allusion to the floorboard. Even something as simple as her telling him in a letter about it would have created enough context for the watcher to know he was aware of it and could later use it. Better still, they could have showed him loosening the board so it squeeked in the first place.
If they did edit out an earlier scene of her complaining about the squeeky board to him in a letter or of him placing the book under the board, they could have just as easily edited out the scene of her finding the book, except that would have left the 'promise' to get it to her hanging in the air. It would seem to me, leaving the book would be a way he could still reach out when she stopped picking up letters. A determined man who was resourceful enough and knew architecture, could figure out that was a way to do it, especially if he knew she already had complained about the squeek. So it was probably a editing choice seeking a best-compromise. 45