Well, it sounds like it could be something to look into, and now would be a good time to start a conversation with her teacher because any move is likely, in my experience, to take months. So what you may more realistically be looking at is a better fit in the fall.
I would start the conversation, like I said, with the current teacher. What has the teacher noticed about your daughter and the educational fit? Sometimes the teacher will see that a child is way ahead and will work to provide her with challenges, sometimes the teacher will notice a child is way ahead and will simply think, "that's great" but make no move to adjust the work, and other times the teacher will simply have no idea how far ahead your child is. If you have any stories or evidence that show what you are seeing, share that with the teacher. If the teacher is skeptical or wanting more evidence of you DD's advanced abilities, ask for testing, like MAP testing, end-of-year testing for the current grade and for the next grade up (and so on to try to show your DD's actual level), and abilitity testing (IQ). That way you will have factual information from which to work. The principal and school psychologist will probably get involved at this point, and possibly the Gifted program coordinator if necessary. If the teacher completely dismisses you, you will have to go to the principal as your next step.
It is always good to go in with the attitude of gratefulness for what they have done so far and a spirit of cooperation to figure out the best situation for you DD. Do no go in demanding anything, and try to keep things pleasant. Help them to see what you see and keep an open mind about what they say they see. What you are looking for is to begin the process. Nothing will get resolved in one meeting.
It would be worth considering, IMO, why your DD seems content and whether she actually is. It is fairly common for girls, in particular, to hide their academic abilities because fitting in is so important to them. Not saying that your DD is doing this, but it is worth considering. Of course, if she truly is content, that is important information to consider as well.
