I wonder if the double touch of the home button that pulls the screen down should have something extra added to it?

The blank space up top could be filled a mini lockscreen, allowing access to the continuuity button from anywhere.

Recently missed notifications could also appear in this space, easily swipable to their app as they are on the lock screen.

There could also be a way to return to the lockscreen from here, such as with a drag gesture down.

All this functionality would be brought in by the home button. An accidental Touch ID unlock could segue back to the lockscreen’s controls, all from the thumb.