It's not really that any tube gives you more headroom at all.
It's that the 12AU7 (or whichever) is less sensitivity.
It breaks up later, rather than earlier.
The higher sensitivity tube like 12AX7 breaks up earlier. It is the most sensitive of all of them.
Gain: they all have the same gain.
The gain is limited by the power supply voltage.
The 12AX7 more sensitive, reaches limit of gain quicker. Breaks up quicker.
The 12AU7 reaches limit of gain later.
But both tubes will produce the same amount of gain before clipping, limited by the power supply.
So once again, "gain" and "sensitivity" are two different things.
Frequently mixed up, or called "the same thing," which they are not.
The only way you are going to increase the actual headroom, is to increase the power supply voltage.
If you increase the power supply voltage then:
you will have more gain, before limit of gain is reached.
More available voltage = more clean headroom before clipping.