I do not know SongWriter, but I suppose that you can set up a new document’s staff order in the Document Setup Wizard.

Then, copy the music over from the old document.

I don't have Songwriter nor can I make the manual display properly. Usually, this is a setup issue. When you go to the setup wizard, select Lead Sheet. This should have the staves ordered properly.

If it's something like NotePad, your setup wizard will be something like this.

See how I have Tenor Voice highlighted. I can use the scroll arrows to the right to move it above or below the Piano. That's the order it will display on your staff sheet.