Adding wood during sear in weber kettle?

I had a cheap gas grill that was garbage for searing steaks so I picked up a Weber kettle grill yesterday. I'm cooking my first steaks tonight using the 2 zone reverse sear method. I was pondering whether to add some wood right before searing. Is this a thing, does it add a hint of smokiness? What kind of wood is best? Could I add wood during the whole cooking process? Thanks! via /r/steak https://ift.tt/2JrZciG