This is something that tripped me up when I was streaming Majora's Mask yesterday for the 6 Day/18 Mask Run practice. Basically, if you don't talk to the Postman when he stops at Clock Town's east exit before 5 AM to get the Postman's Hat, he will leave naturally. Because of that, I ended up losing out on the Postman's Hat. So my big reward for even killing all 4 temple bosses before it was too late was an actual migrane that I had to deal with the rest of the night, so I ended up falling asleep before Toonami even came on.
Given my review of the Postman's behavior along the way, however, I suspect that the time spent in Sakon's Hideout does contribute to his movement, despite it already going to be spent on the clock reset to Night 3 Midnight. It would explain the weird timing in his stopping in the 6 Day/17 Mask Run videos. (He takes 2 game hours to get there from when you give him the letter. I got the Postman's Hat a few game minutes before 12:30 AM.) If this is the case, I can stall in Sakon's Hideout and make sure that the Postman is ready to give the Postman's Hat at Midnight on the dot.