Agree with PPers. Acceleration can be very helpful, but you need to look at the child and be willing to adjust when things seem to need further adjustment.
ODS started a math-only acceleration in second grade, when he was 7. School skipped him one grade, then two after realizing one wasn't enough (within a matter of weeks). It helped tremendously. Being closer to his level material wise, even if he was still at the head of the class, was a better fit. He also had some "pull out" project time. He started working ahead in ELA later that year mostly on his own (at his desk or a table). Mid-third grade, he was skipped into ELA two grades up for class-time. It has worked pretty well, although the first term of this school year taught us that he wasn't quite prepared for the expectations of getting everything tracked and turned in on a daily basis. I'm glad to say he's doing very well at that now.
He's benefitted from the blend of age and closer-to-academic-level class mates. Out of school activities are mostly age based. He is certainly asynchronous. :-) I should mention that the school has been willing to do a full acceleration (or probably two) but he has been very opposed to that at this point. It's not perfect, but it works for now.
We have no gifted-only school options and for him, I would have to be convinced that a gifted school would recognize just how quickly he can learn and how little patience any of us have for busywork. By busywork, I mean excess work that is largely designed to add volume, versus refinement or depth.
