You can fire all of the former staff's assistant coaches. That is standard operating procedure in a coaching change, and in fact *should* happen (and should have included firing the interim head coach instead of making him the new offensive coordinator!).
What you shouldn't do is ALSO fire every single other operations staffer, recruiting staffer, high school relations staffer, and player personnel staffer in the building before you have an opportunity to make your own hires or give those folks an opportunity to audition for the job. Yet that is exactly what Belichick and Lombardi did. Heck, they even axed the longtime executive administrative assistants. It's actually extremely unheard of to completely clear out an entire football staff like they did, and is a huge reason why the first few months were a disaster. The vast majority of the non-coaching personnel that were fired have been here spanning several coaching eras- some had been here since the Bunting days. Made no sense to fire all of those people on day one until a plan was in place to either retain or replace them, and the way that we can know that Belichick and Co. recognized that it was a big blunder on their part was that they went begging to several of those folks for them to return on a temp basis- most told them to GFY.
Fair disclaimer: I remained close with several of those folks whom I'm referring to long after my days of being affiliated with the football program, so I'm certainly not without my biases in being mad at how those folks were treated. I understand it's "just business" but it was stupid, ineffective business, IMO. Think of most of those folks as being similar to the non-partisan/non-political people that get needlessly fired by presidential transition teams.