Going bright red

I`ve been having my natural light brown hair (level 5 with strong red/gold undertones) double processed to acheive a light cool blonde for a few years now (if I try to do it in one step, my hair goes seriously brassy) and have been interested in going a true red colour.

I went to my colourist the other day and she filled the bleached ends. She then put a light red shade with a touch of pure red colour and 20 volume peroxide on the regrowth and 20 minutes later did the same on the bleached ends. Unfortunately, the regrowth didn`t lighten much at all and pretty much looks brownish with a strong red undertone. The colour on the bleached ends looks much better and more like the colour I wanted. Of course, because they have been bleached, the colour is fading out quite fast, but she said that this wouldn`t happen as quickly after I`ve coloured the hair a few times.

So now I`m left with brownish-red roots and brighter red ends. Will I need to continue to bleach the roots to acheive a true red or can this be done with a high volume of peroxide instead? She insists that we shouldn`t bleach because red fades fast as it is and bleaching will only make the fading worse.

I`m including a link to a picture of the hair colour I want and was hoping that someone out there might be so kind as to give me their own opinion on what to do. :)

Thanks!
Melissa