I had one who stayed at their school. They were happy, got the grades and they did the subjects they wanted to study. Got great A levels and are now at a top uni. Easy
I had one who moved from their outstanding state school, same one as their sibling, to a Co-ed private school. Main driver was that they had never really liked the secondary school so had decided by year 10 they would want to change. The state school didn't offer the subject they wanted anyway so that made the decision.
We looked at:
another state school similar to the current school where they knew people,
a really great state school with a huge sixth form where they didn't know anyone
A highly academic state 6th form college which was a pain of a journey and where they wouldn't be a priority candidate due to location, being highly selective and being in a school with a sixth form
A very progressive co-ed private school
Another co-ed private school.
We discounted any single sex schools, state and private also any with 6th form entrance exams as they didn't want to do them but was happy to do interviews and spend a day there.
In the end we had offers from everywhere except the 6th form college, we discounted the progressive private school as the deposit was too big to lose if it wasn't a firm choice. We held both other state and the one private place until results. In the end one state school didn't have enough demand for one of the options so didn't run it and so that was off the list.
Then child decided the other state school was too big and couldn't see themself there so we were left with the co-ed private school,
Now year 13 and best decision ever but it wasn't made by us but by our child and to be honest, at 6th form you can't do it any other way.