Maybe the past predicts something about the courses of employment of former coaches.
The late former coach Fred Taylor took a job outside the football program. I don't remember whether it was outside the athletic department or not. IIRC his duties included (1) showing up at alumni functions and shaking hands, and (2) acting as a paperweight (no personal computers in those days; we used lots more paper than most people use today).
Seriously, he may have had something to do in fundraising, but he was definitely not front and center as a fundraiser.
I don't remember any of the few football players that were personal acquaintances of mine ever saying anything uncomplimentary or unkind about Coach Taylor (other than the mild grousing about coaches that I suspect most college athletes do). Some of his assistant coaches and other functionaries in the football program, on the other hand, were soundly disliked. Stories of bad behavior by those fellows began to circulate after their TCU employment was terminated and they were no longer in a position to retaliate against tellers of tales.
In those days, football head coaches' compensation was this side of the stratosphere, and Coach Taylor possibly wanted a post-football job to keep his household lights on. AND, IIRC Coach Taylor was a tenured faculty member, and TCU may therefore have owed him employment of some sort right up to retirement.
I was a student in times around and near those happenings, really at the end of old-time TCU football that was last great in the Abe Martin era. TCU attempted a modern re-start with Coach Pittman, and most of us know how that worked out at least in the pre-Franchione days.
David McWilliams is a fired UTx football head coach that stuck around his university, post-football. He is or was (don't know whether he's retired these days) a damn good fund-raiser. One of those people who, face-to-face, is hard not to like. IIRC he was one of those people that was essentially on campus not only for college but for his whole professional career (save a few years at TTech, after the DKR staff was let go), so he knew everyone that was anyone at UTx including BMDs. He is/was one of those rare people that get along with everyone (except maybe boosters that wanted better football than he provided as head coach).
So fired football coaches, as opposed to coaches moved upstairs after retiring from a glorious coaching career, sticking around seems to be a very rare phenomenon, tied to a narrow range of circumstances and personalities. I doubt that Coach Patterson needs a job for financial reasons, and he might not be someone that everyone that is anyone at TCU knows and loves. And TCU has an athletic director, so there's no place to kick him upstairs gracefully if that is even possible after a mid-season bail. I'll be surprised but not sad if Coach Patterson does not put TCU behind him.