Unfortunately, I wasn't able to experiment with a different site or improving things more because we only had about 30 minutes without rain.
The reflector really helps with the shadows on the lefthand side of the photo. The darkest shadow is not completely black, the cloth looks smoother, and the colors show more than the texture. And overall, the garment is slightly brighter than the background.
There is still a problem on the right, though. I noticed it at the time, but didn't know what to do. Now it's obvious - turn the dress form to get light into that shadow. I'll try it again if we get a break from the rain tomorrow. It's POURING right now after sprinkling all day so I'm not holding my breath.