I recently discovered some of the shops that sell cigars have scraps of thin veneer that comes in the Oliva cigar boxes. Cigar smokers use them to light their cigars avoiding a chemical taste. I grabbed a few to use on my head stocks to hide the (scarf) joint on top. I use water based contact cement to glue them on. Then I glue my side pieces so they are the same depth. I also used one on a factory neck to hide the manufacturer name.
Make your scarf joint in the opposite direction, the joint will be under the fret board.