To unlock the One More Time trophy in Spider-Man, you must finish the game a second time using the New Game + feature added in Update 1.08.
You cannot start a New Game + unless you have a save file in which the main story has been completed. You can then create a new game save file in the main menu. You can adjust the difficulty and other settings once the save has been created.
In New Game + your unlocked skills, suits, mods and gadgets carry over from your completed save as well as your XP level. As with a regular save, the level cap remains at 50 but each time you level up after hitting 50 you get an additional 1% health and 1% melee damage.
As this trophy necessitates a second play through you might as well combine the effort with the other new trophy which requires you to complete the game on the new Ultimate difficulty -- this way you can start the hardest difficulty with a significant advantage.