Is there any red hair dye that could dye over dyed black hair without the use of bleach?

If your hair is black, there is no way to get it any other color without lightening or lifting first. You don't have to use bleach to lighten your hair, but you will have to use some sort of chemical lifting product. There is no way to completely avoid damage, but you can minimize it. Since the black in your hair is a dye, you can look into using a color remover, which strips artificial dye from the hair, and is much more gentle than using bleach. You can buy this from a professional hair supply store. Lifting haircolor is a very delicate process, and multiple factors can affect the end result such as any previous chemical colors or processes, the porosity and general condition of the hair, the natural color, the underlying pigmentation, etc. The hair does not lighten uniformly, and can take multiple applications to achieve the desired level of decolorization; however, it is not suggested that you attempt to apply any color lifting chemicals more than three times in a one-week period. Red is also a very tough color to keep in the hair. The pigment molecules are very large and have a hard time penetrating, which tends to cause them to fall out very easily. If you have previously lightened your hair and its porosity may be compromised, opt for a porosity equalizing demi-permanent color or a porosity equalizing treatment. You will most likely also need a filler, which will set the pigment base for the red molecules to cling to. After the filler, the color of your choice may be applied, and this will complete the process. The best way to achieve your ideal color is to make an appointment at a hair salon for a color correction or see a color specialist. At the salon, they will have all the products they will need immediately available to them to optimize the process and minimize the damage.