In this situation I usually remove the existing boxes completely and replace them with blue plastic remodeler boxes. These mount to the drywall using wings that fly out when you run the screws in.
By taking out the receptacle and moving the wires aside you can see how the old boxes are mounted. Often there are there are nails passing through the inside of the box. Pry those out or cut them off and gently remove the box. Carefully trim the drywall to fit the new remodeler box, and secure it using the mounting wings. You'll want to pull the cable into the new box before you fit it into the drywall.
This type of remodeler box has limited space for the receptacle's mounting tabs, so you may need to trim the ends of the mounting tabs to fit in the available space. If you don't do this they can keep your trim plate from laying flat against the wall.