Here's what caught my eye:
"There is a broader statistical angle to this effect, as well: If a head coach is going to see a leap, it's probably going to happen pretty quickly after his hire. Over the past 10 seasons, 23 teams have seen their S&P+ rating improve by 18 points per game in a single season. Eighteen of those teams were led by a head coach in either his first, second or third season at the helm. And among that range, the second year is the most likely chance at improvement.
In a coach's first year, his team's average change in S&P+ rating is minus-1.03, with a minus-0.12 average change in win total. By Year 2, the average change is plus-2.59 (plus-0.87 win total). In Year 3, the average change in S&P+ rating is plus-1.44 (plus-0.26 win total).
Over those 10 years, 70% of second-year coaches enjoyed a win total at or above that of their first year and 63% saw their S&P+ rating improve (46% by at least three adjusted points per game, 17% by at least 10)."