Waiting Before Re-dyeing Hair?

I dyed my daughter's hair a medium red on top and dark black on the bottom about 4 months ago. The red has faded to almost an orangish brown and the black to a medium brown. She decided she wanted to try blonde. I bought the Garnier Fructis non bleach blonde. It said it was made for dark brown to blonde color. It is not even close, it barely lightened her hair at all, it is still the same colors. Can I go to the store and buy a blonde with bleach and use that tomorrow or do I need to wait a certain amount of time? Thank you for your help.