When hunting monsters, you have to defeat monsters in a serial order starting from level 1 regardless of your own level.
Even if you are Lv. 20 and the monster you wish to hunt is Lv. 19,
you can’t hunt this monster until you have hunted a monster of Lv. 18.
The attack button for Lv. 19 monsters is activated once you have hunted a Lv. 18 monster.