I agree with Mike. In fact, NFHS Rules specifically state that a coach who has been ejected must leave the vicinity (out of sight and sound) or the field and can have no further contact with the participants in the game until the game is over (no cell phone calls from the parking lot.
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Some offenses require immediate ejection such as wearing metal cleats, malicious contact, and fighting. Others can result in ejection or restriction to the bench based on the umpire's judgement as to the severity of the offense.