This is a tough question because every school/situation is different. We had DS tested privately, submitted the report to the school early on (to the superintendent), and then met with the asst. superintendent and the school princiapl to review the report and talk. They then met DS a couple of weeks later and put him in a K and a first grade classroom for part of the day and decided that he should skip K and go to First, which I am supportive of. I didn't ask or push for it (although it was very clear that was one of the many recommendations from the tester who submitted the report). But we also don't have any gifted coordinator in our district or any gifted programming, so I couldn't have contacted them. I think out situation worked out nicely though.
If you aren't interested in doing the private testing just yet, I also know that most schools are required to do a K screening. In our state (NY) they are required to do this screening and required to let the parents and the superintendent know the results of any children that may be gifted (within 10 days). Unfortunately in our state that is where it ends, I don't think they do anything else with the info. Screenings are very basic (colors, shapes, letters, numbers, etc.). Our DS didn't actually take it this year because they said it wouldn't be necessary based on all his testing and info they had on in. At least if they did that testing though, the school would have some idea of what your concerns are. Also, it may be helpful to just have the pre-k teacher write something up and then send it to them and see if you can schedule a meeting or speak with them about it. Maybe check to see if they can do some sort of testing in the fall.
Good luck! Keep us updated!