Right, you need to be careful putting "mid brown" over bleach highlights, you can go a bit green.
I hear ya about the multi tones, though, that is lovely isn't it, does make a difference.
If your natural hair colour can be "reddish in the sun" then you have natural warm tones and I think you should strand test a colour called Clairol Nice n Easy Natural Medium Golden Brown (117d I think). This does not have enough ash in it to turn you green, but is warm in a gold kind of way, not a red kind of way, so you shouldn't go ginger. Look at my profile pics, you will see it with captions explaining it. I did it over blonde highlights myself just this year. I "filled in" with Rimini Lighest Brown (Loreal) first but I may not have needed to do that step. Only a strand test will ensure you don't go too "warm" if you do it over blonde though.
To do a strand test, snip a small segment of hair (really small, about 10 strands) from an under-layer of hair (try and choose your lightest colour though for your strand test) - it doesn't even need to be a long strand, secure it with a sticky label (or sticky tape but a label is better for some reason!). Mix a teaspoon of the developer and a teaspoon of the colour from the box of dye together in a ceramic bowl. Use a plastic spoon if you can. Place the strand test over some kitchen foil, paint the dye over it using a (washed) foundation brush (or similar) and fold the foil up like a little packet, leave for development time (about half an hour, read the instructions). Put it in a warm place (airing cupboard, under your laptop, or down ya bra!)
Wash it off after development time, dry it, and observe it in daylight and preferably strong sunlight before you decide to colour your hair that colour.
Hope that helps x