That's the problem - there is only one school tag.
I don't see that as a problem. Most of the stories I've seen with the school code have the character start in one school and move onto another. Typically they start in the local school prior to high school, then move on to high school before going to college. At present the school code covers that, but if you break it up you then have a serious issue on how many codes and how they apply.
Note I'm ignoring everything prior to K1.
Let's take the school system as being K 1 to K 12 then College. In the K series you get some schools that do K 1 to K 6 while others do K 1 to K 4, and others will do K 1 to K 5 - varies between states, systems, and countries. The names will vary as Grade School, Middle School, Primary School etc. Then you can have K 5 to K 8 or K 7 to K 9 or K 7 to K 12 or K10 to K12 or K 11 to K 12 with names like Middle School, High School, Senior High School High School, College, Finishing School. Next is College and University which can be from 3 years to 5 years for the under grad degree and then tack on more for Honors or Masters then Doctor School and other specialist school. Next you can add in vocational school and other technical school.
In the state where I live we have schools that do K 1 to K 6, others do K 7 to K 12, while some do K 1 to K 12, and some do K 7 to K10, other do K 11 to K 12, we have technical colleges, universities, military colleges and some that do K 11 to the 2nd year of a university degree.
The typical US based school story starts with a kid in grade school or middle school and follows them through to high school then college. If you split it as being three codes as K 1 to K 6, K 7 to K 12, College / University, then everything else such as story as above would need all three code, and maybe 4 if they also do a tech course.
It all sounds like too much hair splitting to me.
We already have issues with the hair splitting on the incest codes where some think you should only tag incest while others say you should tag incest and the relationships to get a family fun night having the4 codes of Incest, mother, son, daughter, father, brother, sister instead of just incest. The same issue occurs with the BDSM and Interracial codes. Authors differ in the opinions too. Where there are multiple codes that can be used for the same aspect of a story you'll have some authors use one, others use another, and some use both. If there are 10,000 stories with the school code you might have grounds to split them up, but there's only 1,880 of them and when you include other categories or exclude other categories, you get a lower number. Codes +school, -harem, -polygamy/polyamory returns 1,754 stories; +school, +teenagers gets only 335 results; exclude the codes with adults and it's down to 103 stories. Vary the sex content and other tags to refine it further if you want.