On this page of the guide to Disco Elysium, you will find a walkthrough for The smoker on the balcony quest.
Your objective in The smoker on the balcony quest in Disco Elysium is to talk to a mysterious smoker who can be met on the balcony of an apartment located in the second part of the courtyard, behind the Whirling-In-Rags hotel.
Go to the courtyard behind the hotel and talk to the man standing on the apartment's balcony. You'll find out that you're not the first to ask him about the hanged man - a muscular guy came to him and asked the exact same thing. If you pass the Suggestion skill check, the smoker will agree to talk to you again.
The man will point the stone lying outside the door of the building. Look under it, and you'll find the Key to Apartment Building.
The task can be done only after 9:00 pm. Unfortunately, the smoker will not help you solve the murder case. However, he will direct you to his friend (Sunday friend), which will start a new quest – Interview the Sunday friend.