What kind of applique are you doing? Fusible, machine with turned-under edges, or hand applique with turned-under edges?
If fusible, I would fuse two pieces of the white together before cutting out the snowman.
If invisible machine applique with turned-under edges, there are a couple of possibilities. If it's freezer paper applique a la Harriet Hargrave, then I would applique the snowman as usual, cut away the backing fabric, and insert another cut-out snowman underneath, spray-basting to keep it in place. Be sure it's a dense enough white piece of fabric (doesn't have to be the same as the snowman).
If machine applique with satin stitch edging, either of the above methods would work.
If doing hand applique, I would cut out the white liner a little smaller than the original snowman and baste the two pieces together before appliqueing.
Edit: Oh! I like the idea of fusing an interfacing to the snowman too.