I agree with your diagnosis: defending the first header is not the end of the set piece.
Before the corner is taken, I want one or two players responsible for what happens outside the box. One may be your striker or wide player positioned to pressure the clearance receiver, with another ready to squeeze onto the second ball.
The back line also has to come out together once the ball is cleared. If six players remain on the six-yard line, the opposition have time to recycle.
I’d practise second phases deliberately. Start with a corner, then have a coach feed a second ball to the edge or wide area whether the first delivery is cleared or not.
The target is not sprinting out blindly.
Can the first player pressure the receiver while the rest of the team moves the line away from goal?
A set piece often lasts 15–20 seconds longer than the first contact.
Coach that whole period.
