You asked about ways of improving state schools without throwing money at the problem, OP. Well, job one is to solve the recruitment and retention crisis.
Firstly, do something drastic about teacher workload. The best solution would be to give teachers less contact time, but you can't do that without employing more teachers to cover the shortfall. Not doable without more money, and impossible anyway atm as there simply aren't enough people who want to be teachers, even with a big bursary to train).
So... put schools back into LEA control, ditch Ofsted, get rid of league tables, ban written homework in KS3, set national guidelines for much stricter behaviour management (SLT to routinely remove troublemakers from lessons and supervise them) and stop deterring schools from excluding students, set a national limit of 2 data collection points per year (except public exam years).
Make schools stop making teachers accountable for things beyond their control. Place some of the expectation back on students (who currently know their teachers are more stressed about the results than they are). Stop expecting teachers to do lunch time interventions for students who cba to pay attention in class time.
Also, total ban on prescriptive uniform rules. Only generic e.g. navy trousers/skirt, white shirt etc. Allow trainers and whatever coloured socks. Basically stop making petty rules which waste teachers' time and piss off students. Ideally ditch uniform altogether. And I'm just getting started.