There are now a wide variety of cheap floor lamps available, some of which have both an upward-pointing main light and a downward-pointing reading light. Unfortunately, the plastic shades can't take the heat trapped in the downward-pointing reading light, and essentially disintegrate. My improved replacement closely copies the shape of the original shade, but with the important difference that it has vent holes to allow the heat to escape, as you can see in the photos.

I can't guarantee that this shade will work for you, but I'm pretty sure replacing the original shade with an identical one wouldn't work.