We are just starting on this road, but I am also a teacher, so I have some idea how things have gone on that side of the coin as well.
My second grade son will be transitioning into 3rd grade reading and math classes, while remaining in his second grade for the rest . . . mostly the social and specialist aspects. He is academically above even a third grade level in both math and reading (and others), however it is a good place to start the process, in my opinion. The plan is, assuming he continues at this pace and is ready for and okay with it, to move him into fourth grade at the start of next year, potentially going fifth grade math and reading after winter break again, and finally moving into sixth grade in what would normally be his fourth grade year. This is middle school for us and the options are much more available in terms of acceleration and opportunities. (all that said, we don't know what path he'll take until we get there . . . time will tell.)
I mention this because it might be worth bringing up at your school? Is it possible to accelerate some classes over and above what has already been done?
We have an administrator in our district who is new this year, but who has had ample experience with acceleration in her children's home district. We were fortunate enough to have her attend a meeting with our principal, our son's teacher, my husband, and myself simply as support all around. The thing that she said that stands out to me (and really makes sense from a teacher-view as well) is that it isn't the content, but the pace, that sets these kiddos apart. We can accelerate them in school, but generally, at least at the elementary level and in most classrooms, that is only a temporary solution.