Coach contracts are much simpler than player contracts. You can sign a coach during any stage that you can sign a player. You can sign an unemployed coach at any time; you can also promote one of your coaches at any time. Signing a coach will automatically fire your coach at that position. You cannot fire a coach without signing a coach to take his place. If you enter a game stage and you are missing coaches, the game will sign a coach for you.
During early free agency, you can hire a coach from another team as long as you are offering him a promotion (position coach to coordinator or head coach, or coordinator to head coach). Outside of early free agency, you cannot hire a coach from another team.
Your available cash flow for coaches is determined by your attendance the previous year. The range is not significant, so if you have a poor team you should still be able to make reasonable offers for coaches.
The Head Coach determines your available playbook. For each player, his coaching hierarchy above him determines how quickly his attributes improve. His position coach is weighed at about 20%, his coordinator is weighed at about 30%, and his head coach is weighed at about 50%.