He calls the CEO to ask a "personal question" so to skip the assistant, asks something innocent, then let's the CEO he has a new number and provides a fake number. He asks the CEO to confirm he heard the number correctly, but it's a bad line, so speak clearly please.
The "new phone number" has all the digits of the bank account he's trying to hack. The account is likely the account number that he's being paid for the consultancy work with. He could have got this simply by asking to confirm from which account he'd be paid from to confirm the transaction.
He is asked to report his review of the new security system to the board (given it was a large investment by the Bank, or just the wrong word used) and the architect would of course be invited to his own project's review?
The board then asked Mitnick to design a new system and said that cost wouldn't be an issue.
That all seems pretty easy to put together?