I am a supply teacher. Firstly for longer term contracts, you don't always get a good daily rate as the supply agency takes an awfully big cut so I have covered a class and worked out that my pay was below minimum wage. After 12 weeks I am entitled to be paid to scale and it's only then that the extra workload and responsibility for long term supply is worth it.
Is the class particularly difficult? I'm often requested by schools but sometimes the behaviour is so difficult and if it feels as though there is no back up from slt, I will not stay. This isn't fair on the children but it is a fact that will have an impact on a teacher's decision to stay or not.
If a school treats me well and the staff are nice, I will put myself out to help them.
Unfortunately, at the moment Covid will be causing so many more problems. At the moment I only want to work in one school to reduce my risks. I don't want to be moving around and despite my agency's pleas, I am turning down work at the moment as I am in my 50s and work with ks1 where social distancing cannot happen.
Many supply teachers just don't want to work full time due to caring responsibilities. This is where a school will perhaps have more succes getting 2 teachers to job share. I have done this many times and it means I am more likely to stick at the job as I don't need to take time off for taking my parents to hospital and gp appointments. Maybe this approach would be better.
The school will be trying their best to make sure your daughter has a stable teacher next year, that is a given and doesn't need to even be mentioned. Talk to the school about your daughter's stress though, including any current teachers she has. They will want to help.