The problem I see with wood that's already oiled is that it may interfere with the glue adhesion. Put a drop of water on the wood and see if it soaks in and changes the color of the wood, or does it just sit there. If it soaks in, you can use Titebond or similar. If it doesn't, you'll need to use polyurethane or epoxy. Messy messy . . .
“We should be careful to get out of an experience only the wisdom that is in it and stop there lest we be like the cat that sits down on a hot stove lid. She will never sit down on a hot stove lid again and that is well but also she will never sit down on a cold one anymore.” - Mark Twain