That is exactly what i did. U won't separate it where it is secure.
Didn't want to damage it,..
Didn't clean the old glue either,
Just applied the Contact cement to the old glue where i could get to it.
It is almost exact as original cement.
Used spring cloth pins to clamp till dried.
Don't use any tape on woodgrain it will pull it off,
Near disaster when I taped the template to paint the brown circles.