The problem you have is that you are going to be classed as a late application and therefore will only be considered after every body else has been given a place. As such you might not have any choice in the matter, it will be a matter of which schools still has places available.
Our local schools are ok but not particularly high achieving. We knew if DS went there he would coast. Us pushing him wasn't going to get the job. He needed to be in an environment where he is with kids brighter than him. Only then would he push himself to work harder.
If your DC is anything like mine then I suggest choosing the most academic school you have access to.
Having applied to secondary school for DD (yr6) out of area as we were moving "back home" with a job relocation, it now looks like we are staying put. DD's secondary school choice was all arranged and we were very happy with it, great all girls school that she had a good chance of getting into.
Now I have 5 days to make my mind up in an area that I don't know very well. The choice is this, we live in small town with a much improving academy, good local reputation, small school, 700 pupils, 11-16yrs. Alternative is a large, 1800 pupils, high achieving (top 30 in the country) CofE secondary. We can probably get her in as we will have enough "church points". This school is 30mins away on public transport.
The majority of her small primary school class will go to the academy, the rest being split between various grammar, private, catholic schools in the area. I have no idea if it is best to keep her local, then let her go further afield for post 16 or just go for it now. She is bright but needs pushing! Any advice very welcome. Thanks.