Adding some poultry netting might work until the snow flies....but even my 14ga 2x4 welded wire run roof, supported at 6' spans, grabs wet snow and starts to seriously sag.It's a 4 ft fence, and if we have issues I think we'll be covering it with some poultry netting to keep them in. If that starts to happen, we'll probably have to change plans and add a run!
Keep snow loads and clearing in mind with all your plans/mods.