I've used an auburn hair dye for ages, having naturally brown hair, (going a bit grey). I recently started to find it rather harsh and put a Nice N Easy Natural Dark Champagne Blond (106) colour on it.
Was terrified it would turn out dreadful and was resigned to walking around with a paper bag on my head but I'm delighted with the results. All the reddish tints have gone and my hair is now a light brown/ fair colour and very natural looking. However, if you've had your hair dyed recently, another dye so soon might be rather harsh. If you use lots of conditioner after every wash and go easy with a hot hair dryer hopefully it will be OK. I didn't find any shampoo which effectively got rid of the red/auburn colour previously, although exposure to lots of bright sunshine did bleach it our and turn it fairer.