There are many ways to remove red tones from hair without having to bleach it. Here are some of them:

1. Dye it an opposite color. Green or cool ash dye will cancel out hair that’s dyed red. If your dyed hair is on the warm orange side, use a completely blue hair dye.
2. Use a toning shampoo or conditioner.
3. Strip the color using baking soda or vitamin C as a paste and applying it hair in the shower.
4. Further bleach or add a color-correcting dye to your hair.