Semi Permanent color put over Permanent Color

Hi! My hair is professionally colored medium auburn. My natural color is dark brown, almost black. I want to brighten up the color, but I would like to do it myself if possible, with a semi-permanent red color placed over the existing color that I have. My goal is to basically be brighter and redder. Is this something that I can achieve successfully on my own by simply applying a semi over the color I already have? If so, do you know of a brand that would be the best to use and has nice, rich red shades?

Thanks!