As you noted, when your Health drops below 0 (zero) -- considered Hurt -- you must make a Consciousness Roll although you'll automatically succeed if you've only dropped to -1.
If you suffer further harm and see your Health drop to -6 or lower -- considered Seriously Wounded -- you must make another Consciousness Roll.
According to the 2nd edition of The Esoterrorists (pp 27-28), the most recent iteration of the Gumshoe rules: if you go to Hurt and then to Seriously Wounded as a result of two separate attacks, you make two separate Consciousness tests. If you jump straight to Seriously Wounded, you make a single test, even if this is as a result of spending pool points when you attempt to avoid losing consciousness when Hurt.
As written, you do not make further rolls due to incremental loss of Health within these two ranges. Once you've resisted the physical urge to pass out, you're good. However, as a GM, I think you have the right to call for further rolls if the nature of the trauma means it makes sense.
You might wait for another Consciousness Roll at Health -6 if the loss of Health comes solely from the slow loss of blood (simulating the character going into shock). If the loss comes from something more significant and traumatic, you might well request an additional check. Really, a character with Health below 0 should be staying away from combat, not wading further into it!