White noise----don't really know what that is(the science behind it), heard of that before, a sound that counteracts another sound, think about it, 2 sounds hitting your ears.
But more curious about what you re calling "double plugging?
What is that?
Let me break my opinion down for you.
Minimum for firing 22long rifle rimfires---foam plugs.
But honestly I prefer plugs and muffs for all target shooting,, When hunting , at least foam plugs for rimfires, Muffs for centerfire rounds.
Understand, ANY hearing loss is permanent, once lost ya can NEVER get it back !