Late summer is a good time to over seed the lawn if you need new grass, though you can also do it now if you are committed to regular watering (mornings best to encourage deeper roots).
Depending on the state and size of the lawn, I would consider hiring a rotivator and break up the clay and work some organic matter into the soil (farmyard manure being my favourite) before raking, rolling and adding a layer of topsoil and seed. That should decompact it and help it to retain moisture as well as fixing any lumps.
Once you’ve got new grass, you can add a bit of extra topsoil into any hollows to even it out, depending on how well you prepared it before. Note the seed you used or keep some extra for over seeding in spring.
Personally I wouldn’t bother with turf as with this sun and water grass will germinate in around 10 days.