You check mandatory actions any time you are able to take an action. So when you activate the creature, before you do anything else, you must check. In this case, you have a mandatory action to move "at least one zone closer" during your action phase. Can you do that? If so, you MUST move closer.
After moving closer, you still may take a quick action. So you check again for mandatory actions. You still have the Siren's Call and still have the mandatory action that you must move "at least one zone closer" during your action phase. Can you do that? YOU ALREADY HAVE! Mandate fulfilled, you can now do whatever you want with your action. Congratulations.
@DaveW the order matters in that you're not freed from the mandatory action until it has been fulfilled.