A question about double processing

Double processing means two chemical processes close together, the common one being box bleach (sold as pre-lightener) followed by permanent colour (which is what is recommended on box dyes, which is one reason we don't really like them here lol)
After bleaching or a permanent dye it is best to wait at least a month before any further processing. Gentler processes such as semi/demi permanent oxidative dyes or bleach baths can be spaced a week apart.
Hair can't repair itself once damaged, chemical damage is permanent and cumulative. Time does allow some bonds in the hair to re-form (salt bonds), the pH to level out and the cuticle to settle back down. It also allows your scalp and hair to build up some natural oils, which is crucial if you want to avoid chemical burns.
Waiting a month also allows you to fully assess the true condition of your hair, as bleach damage will take a few weeks to show.

You could ask for your highlights to be re-touched now, but it is a fiddly job separating out only the previously dyed areas so will take time and a good colourist. You could also try toning them to make them less brassy by using a violet dye diluted in conditioner, so it is a bit stronger than the shop bought shampoos. Or apply the shampoo to dry hair and leave it on for much longer to get the maximum toning effect.