If I can't make it to the showers at the state park in which I generally mountain bike before coming home, then I change in the bathroom. I keep a towel and washcloth with me where I can at least wipe the sweat off my body before putting on clean/dry clothing. I've seen women in the parking lot at the same park doing what you describe under a large towel.

I know another woman who purchased a long casual skirt that she uses for the same purpose that you use the towel for - and the advantage is it can't fall off