The defense ministry was vacated with the arrest of its criminal occupant the morning before the coup. The four branches of the military put Pinochet in charge of the whole armed forces that morning because his branch was the most senior.
There was also no constitution, de facto or de jure thanks to Pinochet.
Actually you can thank Allende for that. The constitution was recognized to be in a state of dysfunction at least three weeks prior when the Chamber of Deputies met and found it usurped by Allende. That is why Frei specifically described Pinochet's action as legal.
Please point to teh constitutional provision that allows for the various heads of military branches to name a new defense minister.