Yes, have a look at Tinkers' link and contact Kendrick to be put on their waiting list.
Having had a daughter start school in Reading in y9 (after being HE'd) I've been through your process, and it's largely a matter of which subjects she might want to do for GCSE, and which schools have places - so you need to get the ball rolling asap. We are in central Reading: I looked at every school which was vaguely within commuting distance, and narrowed them down by which did triple science and Spanish gcse's - DD's choices. Came up with 4. Phoned one (in Reading LA) who were unhelpful, didn't want me to visit, and neither they nor Reading council could/would tell me if there were places, a waiting list, etc. Phoned two in Wokingham LA who were both friendly and invited us to visit, and Wokingham LA were helpful as regards availability. The 4th was not easily accessible, out in the West Berkshire countryside, so we didn't bother visiting, but West Berkshire LA were also helpful when I spoke to them later.
She ended up at the one she preferred out of the two Wokingham schools, and is now nearing the end of y10 and doing extremely well there, probably better than her elder sister who's doing gcse's now at Kendrick
Pm me if you want to chat more about it, but basically my advice to you is to look through all the school websites today and get on the phone tomorrow