There is a school in the city (Eynesbury College) which is just a senior college - ie year 11, 12, and 13.
They are not typical high schools - there are no uniforms, no bells, and 2 hour lessons (each subject has two 2 hour lessons per week, and there are 5 subjects in your final year, year 12).
They consistently achieve some of the top marks in the state and have a whole lot of students from around the world and Australia. & they have a whole range of subjects. The teachers are all of a high standard.
May be worth an investigation if your son doesn't want to repeat (note he can do year 12 over two years at these schools), or if he wants a more relaxed schooling time (it's considered a school between high school and uni). The thing about these schools is everyone goes there for the first time when they are 16, 17, or 18 so your son wouldn't feel like he's a newbie amongst established groups of friends.
Otherwise, if you're looking for a public high school, I think you could still find a school that would take your son in year 12. Just note that year 13 is becoming more accepted and common if he wants to try doing year 12 (without repeating year 11), but finds his results aren't as good as he wanted or he just wants to take a half-load.
There is a great site called SA Central (look it up on google if you want to find it) which has a list of all schools in SA for further info.