Are you talking 9" over all height or just the root container?
If its just the root container, then the next thing that limits you is the no external rez part.
Two systems come to mind - kratky method and hempy buckets. I have never tried kratky myself, but it sounds nasty to me just from reading about the setup. Still, there are lots of folks who swear by it.
I really like hempy buckets myself. I have used them to keep mothers alive for months under really bad conditions - too cold and almost no light. I also use them on every grow to sprout seeds to put in all my aeroponic setups. They are stupid easy and zero maintenance and low effort. Just have a container of your preferred nutes ready to water with when the bucket gets light. I highly recommend Jacks 321 for any hydro setups. I do NOT recommend anything organic at all. The nasty bugs just love that stuff in hydro.
I suppose you could do DWC too, but thats one of my least favorite hydro techniques unless you can keep the water cold - which is very hard to do if the bucket is inside the grow space with lights etc. Warm water grows nasty bugs really well. That forces you to use beneficial bacteria to fight it or bleach or H2O2, etc. None of that has ever worked for me with warm water.
Tell us more about your proposed setup.
Edit: simple plan for a hempy bucket. Take any container and punch several small holes aprox 1/4 of the way up from the bottom. Spread them out evenly around the bucket. Fill the bucket with perlite. Add your plant, seed, clone etc, then fill with PHed nute water until it starts running out the holes. When its time to water again, just fill till it starts to run out the holes and stop. Once a week to ten days, poor in extra nute water after it starts to run out the holes. Maybe 10%-20% extra. Thats to flush any built up salts from the perlite. You can use just plain PH water for the flush if you want., then add some PHed nute water.
Alternately, you can just let some extra nute water run out each time you water. Thats how I usually do it. Im too lazy to flush.