I had to watch the final 30 minutes twice, but I'm pretty sure I have it straight now.
1. In the third time loop, Lee Jae-han spoke to Cha Soo-hyeon rather than Park Hae-jung, and Cha warned Lee not to go to the hospital. But she also - and this is critical - cried about how his death had affected her. She said she'd waited for him for 15 years and he "came back dead." Since he was secretly in love with her, that made him determined to survive and keep his promise to come back. It also broke the loop which had already happened twice when he spoke with Park, who didn't remember him during that transmission. Now Lee had the option to do things different, so he had backup follow him the night he went to the hospital, and they were able to prevent his murder. That gave him a chance to reconnect him with Cha and let her know he had to go after Kim Bum-Joo's master, the congressman whose son r*ped the high school girl. It also changed the future of Ahn Chi-soo, who was arrested, and Kim Bun-Joo lost his status as the police superintendent and went on the run. This solved a lot of future problems, including Park being framed for murder and shot.
2. Once Lee survived the attack in the forest, he had a chance to track Kim Bum-Joo to the warehouse where Kim had the bag containing a copy the disk incriminating the Chaebol whose faulty bridge collapsed. They were attacked by goons sent by the congressman, and Kim was killed, but Lee escaped with the bag containing the disk. He gave that disk, along with a note for Park, to Park's mother; and then he hid out at a river where he was found by the goons again. Before they attacked, he updated the note in the back of his case notebook, leaving a message that only Park would understand: the number of Park's home address. So Park went to see his mother, and she gave him the disk. He released the information and got the congressman in trouble, but the congressman still able to send his goons out again to find the disk copy.
3. Lee survived the attack at the river and recuperated at a hospital. After that, he knew he had to go into hiding, but like Park said, Lee wasn't the type to leave his family in the dark about his location. So he contacted his father - who now had the walkie-talkie because it would have been given to him once Lee was dead. That means Lee's father is the future contact Park speculated about, and Park himself saw the proof when he caught the father burning a bus ticket which would have taken him to the hospital to see his son. Lee's the father helped him hide out in the rehabilitation hospital for 15 years and warned him when the goons were going to attack him there, which must mean there was at least one more loop where Lee was killed at the hospital. Finally he worked out a plan to save himself - and reunite with Cha and Park. A few days ahead of time, he texted Cha with a message only she would understand, letting her know that he was in the rehab hospital and in danger. The goons attacked the hospital, but Lee already knows they were coming and had a plan for escape. Plus, thanks to his messages to Cha and Park, they were on their way to help him at the crucial moment. I like to imagine they showed up at a rear exit to the hospital just as Lee stepped out the door.
