Chino explains that for coffee fortune telling, after the coffee is completely drank (finished drinking), fortune telling is done on the remaining patterns.

チノ：

「飲(の)み終(お)わった後(あと)に残(のこ)ったコーヒーの模様(もよう)で占(うらな)うんです」

In this sentence, 終(お)わる is used because finishing a drink is the normal completion of the action of drinking.

Step:

「①飲(の)みおわる」

The first step listed in the background is finish drinking.