Lay one of your boards over the hole. It needs to completely cover the hole with a little overlap and needs to be the same thickness as the wood of your table top; it also needs to have square corners. Your second piece of wood needs to be at least one inch larger on all sides than the other piece. It is a good idea to use the same wood as the top of your table if it is a table in good condition.
Scribe a line around your board using your pencil.
Drill a hole inside each corner of the rectangle you drew so that the outer edge of the drilled hole just touches the corners of the rectangle.
Insert the blade of your jig saw into one of the drilled holes and cut along your lines. Make sure to cut just inside the line
Remove the piece of table containing the hole. File any corners or other areas so that they are straight on the edges of the table top where you removed the piece.
Center the small piece of wood over the larger one so that the large one leaves at least a one inch flange on the edges. Glue them together like this with wood glue and clamp them. Allow to dry overnight.
Drill six 1/16th inch holes through the flange on your glued pieces of wood. You want two on each long side and one on each short side.
Insert your glued pieces of wood through the bottom of the area you removed on the table top. The edges of the piece that goes in the hole should fit flush with the edges of the table top and it should not stick up above the surface of the table. Use your wood file to remove any other areas that do not fit flush.
Drive screws through the flange and into the underside of the table top with your Phillips head screws and drill. Make sure your screws are not too long or they will penetrate the surface of the table.
Sand any areas of the table surface that are not completely flat. You only need to do this if the board you inserted into the hole does not sit level with the surface of the table and should only be done on edges that barely stick above the table so that you have a flat, level surface without ridges.