Put the inside wood, on the outside after you remove it...but I would think that a layer of sheetrock on the outside would do more for reducing noise than another layer of wood.
If you go for the sheetrock, run some pencil beads of silicone adhesive caulk across the sheetrock before you screws it to the wood (don't nail it).
That should do a decent job. Then you can test it for soundproofing.
If you need more soundproofing...add another layer of sheetrock on the outside, and put it "across" the seams of the previous layer...IOW, don't have the seams all lined up for both layers.
You can add even a third layer if you need it. I mean, eventually you will hit the right amount for your needs.
Then just finish the outside with a nice coat of paint.
You could have done all sheetrock instead of wood...but it is what it is now.
You may want to add another layer of wood or something to the door on the outside.
That's where my logic goes.
On the inside, I would staple a layer of polyester felt across the walls...you can buy it by the yard at any fabric store, and it comes a variety of colors.
The felt has a nice thickness to it, and it's also a good tight weave, so the insulation will be well contained.
If you want to get crazy, do two layers of felt.
You will want to put something also on the inside of the door too...it's a big reflective surface. They have that perforated pegboard that would work well, and also add some mass to the door...then you can also put a layer of felt on that too.
Not saying you can't leave the wood and add some treatment...but I think you will have to cover it with the same kind of insulation that's already there, behind the wood. Maybe you prefer to go that route, and then you have to box off for more insulation...which will just eat up the small space you have. That's why I think adding more heft to the outside gives you more options to get the amount of soundproofing you want, without using up the inside space.
It really comes down to how my soundproofing and what volumes you be dealing with inside that booth.