There's another reason to fire early--if the coach (and fans/etc) think a coach is on such a hot seat that he's "coaching for his job", and then he performs, it's a LOT worse getting rid of him.
Think of the 2012 Purdue football team. It became Danny Hope's last year as HC. He was 3-6 through nine games and they were already inscribing the tombstone. The team then rallied to narrow wins against Iowa and Illinois, then beat Indiana by ~20 to reach 6-6 and bowl eligibility.
And then he was fired immediately following the Indiana win, and an interim coach was the HC for the bowl.
It seemed like a dick move, to be honest. If you've made the decision when he's 3-6 that he's gone after the season, fire him at 3-6 and turn the team over to the interim. Not firing him and then watching him celebrate as his team reels off three wins thinking he might save his job is just giving him false hope (pun intended).
So if you believe Holtmann deserves to go after this season (barring a F4 or NC or some crazy level success), it's better to do it sooner than later--if for nothing else, because sneaking into the tournament and getting a R64 win and then losing in the R32 just makes your decision harder.