Well you could use grass seed, but bear in mind you can't really walk on it normally for 2-3 months. It will also look very different from the lawn you have currently, though it will blend in by summer next year.

It's no easy task adding to a grassed area, its very, very difficult to get the levels right, so you either end up with the new grass too low or too high. I think I'd prepare the areas you want to put extra grass on (removing any plants etc), level and tamp down, then leave it to let it settle properly. Think I'd do this preparation end of July, around then. This isn't easy either because inevitably things will germinate in it, but it does give more chance of getting things level when you turf over it in September, by which time it should have settled sufficiently to be able to work out whether the area needs to be adjusted height-wise prior to laying turf.

I agree with Bamboo's advice fully. Having done this for a friend turf was much easier and 'quicker'. just remember to make sure it gets well watered as September can be a warm dry month.

Thanks very much to you both. We'll start at the end of July!