Should you? No.
Are you going to anyway? Yes.
The Asst. Chief isn't holding your packet to make sure you call and prove that you're really interested in the job; ready to throw out your app if he doesn't get the appropriate number of phone calls from you. The *** Chief (sorry *ahem*) Assistant Chief has lots of stuff on his plate that may be more pressing to the department than getting you hired. As a top administrator, he's busy administrating and approving new hires is just one of many things he's probably tasked with.
While it's important you you, it's probably not very important to him. Just something else he has to do.
Of every one hundred men, ten should not even be here. Eighty are nothing but targets. Nine are real fighters, and we are lucky to have them, for they make the battle. Ah, but the One... One of them is a Warrior... He will bring the others back.
"Wrong door, buddy!"
Let no man's ghost say my training failed him.