My hair is already dark brown but I want it just a little more brownish. I think dying it is a no no since its really obvious its fake. I attach some pics of what I am looking for. It should only be noticeable when the sun is hitting it. Maybe some kind of bleach?
You can try to use slow gradual lightener at home. It works very slowly, so you’ll know when to stop without looking obviously dyed.
On brown hair they give reddish tones, so it may be what you are looking for.
Just two examples: