No, you do not want to use the vegetables that make your stock in your soup. Use fresh ones.
I make my chicken stock from roasted chicken bones. Sometimes you can find bags of chicken bones/parts. Roast in oven until VERY brown - then continue with your chicken stock using these bones versus a whole chicken. This method gives a lovely golden color to the broth. Also, if you cut your onions in half and lay the cut side on a girddle/skillet and let cook until VERY browned/blackish then throw in pot, skin and all, this will also add to a beautiful color. I add parsley, other bits of left-over herbs, along with all the standard stuff that goes in making chicken stock, i.e., carrots, celery. I have even used left-over fennel, tomatoes, asparagus stalks, peppercorns, and flattened garlic cloves. Remember, you are flavoring water - don't be shy with your ingredients.
"Count yourself...you ain't so many" - quote from Buck's Daddy