Wind the clock: When the last play had the clock running, and it was stopped because a REFEREE's whistle, not a player's action.
For example,
Incomplete pass, running out of bounds, touchdown: What stopped the clock? The player's actions caused the clock to stop.
Penalty called after a running play in-bounds: The referee whistle caused the clock to stop - otherwise it would keep going.
Penalty called after an incomplete pass: The referee whistle did NOT cause the clock to stop - the incomplete pass (player's actions) did. Therefore, next play starts on snap.
Penalty called after a false start: Was the clock running BEFORE the false start? If so, clock starts on ready. If not, clock starts on SNAP.
From what you say, the white hat wound the clock incorrectly. Maybe either he forgot that the clock stopped in the previous play for a timeout, or he using a mercy-rule in 'keeping the clock running' in a blow-out game. If it was a close game, stop the clock and inform the white-hat of the previous play.
Throwing the hat: Don't do it in high school. This is not an approved NFHS mechanic. For college/pros, it signifies that you saw a player go out of bounds. In college/pros, the penalty flag is only thrown if that player is the first to touch the ball after coming in bounds. NFHS rule is different - the flag is ALWAYS THROWN when the player returns in-bounds (illegally), thus no need for a hat.
