If you read the Rolls-Royce manual on the 250-C20 series engines, I believe there is a relight procedure that states something to the effect of 'within the 1st 10 seconds, just hit the starter button' with the throttle still full open. It's what the autorelight does...
But it's not in the Bell manual for reasons that have always escaped me. It was in the OH-58A manual if my rapidly failing memory hasn't let me down again.