It's really difficult to work out what sort of secondary school will suit when the DC is only 4.
And just applying to a lot will not make any difference to your chances at each because it's not roll of the dice probability, it's a selection procedure.
I think you need to do your basic research in a published guide - that'll give you an overview of what is available (and logistically possible from where you live or could move to). Then perhaps MNetters can help you hone a short list.
Where do you live? Only these schools are quite far apart and its difficult to know which schools to suggest. If you go to a good pre-prep they will advise you on which schools to try for. Most people wouldn't register for 6 senior schools.