Clean out old glue and debris by gently sanding the solid substrate and the underside of the inlay with sandpaper. You may have to fold a small piece of sandpaper so that fits into a small opening in the inlay. Remove the dust by blowing air into the opening.
Apply glue between the inlay and the solid substrate with a cotton swab. For a tiny space, use a toothpick to apply the glue.
Put pressure on the inlay with your fingers or a block of wood to squeeze excess glue out of the joint. Clean the excess glue off the surface with a barely dampened sponge.
Clamp the repair under a block of wood to apply strong, even pressure and to create a strong bond. Wax paper between the repair and the wood block prevents them from becoming stuck to each other. Allow to dry overnight.
Remove the clamps the next day and remove excess glue by scraping it with a sharp edge. The surface is now ready to wax or sand and finish, according to your preference.