Things have changed in the registration department of avast free, you aren't meant to need to register. But avast are making a right pigs ear of it, considering they say you don't need to register.
By all accounts when it does expire, you are still covered and protected and after ten days it should just renew, a pigs ear or a dog's breakfast, but certainly not a simple registration required.
The main UI interface should change from showing Upgrade to display Activate. That can throw up a confusing screen to select what protection level you want. But that page looks like the only active button is to upgrade, to the left of the screen it should have stick with the free version.
You can also try the Menu option (top right) of the main UI screen and then My Licenses (see attached image1) and that should lead to 'Licenses on this PC' (see attached image2) this will show your license info, in this case still active, but I'm hoping that you will be able to activate/renew from there.