You could try making a soakaway at the lowest point and lead your French drains into it. You would need to dig a large and deep hole and fill it with smallish stones. If you have lots of space would it be possible to drain into a pond or bog garden?

You can also try to lighten your clay by digging in lots of coarse grit and /or very coarse sand, plus as much compost as you can lay your hands on.

As per Steragrams advice. I would use a folk, not a spade, and go over and dig to at least 1 foots depth to loosen up the soil. Compacted clay causes the waterlogging. You will see a big improvement just by doing that.

Organic material, such as compost, will also be necessary to keep the clay from going back the way it was. It's a project, but a meter-deep test hole or two could give you invaluable information about how your soil changes with depth. Sometimes you get lucky in having a well drained subsoil, which simplifies drainage remedies greatly.

We once had a garden on deep clay and found that, no matter how deep a soakaway my OH dug, it just filled up like a tank and didn't drain at all. Good luck - I hope you find a better drained subsoil than we did. If you find the soakaway just fills then overflows, like we did, you learn to live with a boggy garden.

i had the same problem when i moved to my home 11 yrs ago and have found the best solution is to plant large water guzzling plants which make loads of wonderful humus rich soil the area around the Pampas grass and shrub borders is now wonderfully rich and the clay has completely dissapeared so its a long term project