My guess is that when they first met when they were younger, the Director gave Yiyool his school jacket as well as the winter jacket Yiyool had asked for. The panel in chapter 22 shows him taking off his jacket and saying "you want this too?" after saying he wanted to hang again. I'm assuming the "too" means the school jacket? It doesn't show him wearing it, but you could assume it was under his winter jacket. Both jackets were probably in his belongings he left with his friend when staying at their house, so the friend shipped everything over to Yiyool.
Orrrr the author made a mistake and forgot what jacket Yiyool was asking the Director for lmao. Either way, Yiyool had his jacket, and his friend sent the jacket along with Yiyool's other stuff.
I love this sm, but I just have one question, WHY THE HELL WAS THE DIRECTOR'S SCHOOL UNIFORM SHIPPED TO THE MC!! Like how??? I don't get it