Referencing only the Generic Cortex Rules, it seems that damage effects are applied as they occur according to initiative - there is no distinct phase separate from the resolution stage of combat wherein conditions or effects from the action of the round are applied.
Your quote, from page 97, indicates that if damage of any type (Basic, Stun, Wound) exceeds Life Points an Endurance roll must be made. This is presaged by the references on page 12 and 19.
This consciousness roll would occur as a result of taking that damage. If the roll is failed, the character passes out. If the character succeeds, the difficulty to retain consciousness increases each turn, with predictable results.
As the roll is called as a result of there being a level of Stun Damage (as Wound Damage has other more serious effects to resolve first) in excess of Life Points, additional Stun beyond this point does not need to call for additional rolls. The condition of being over the Life Point threshold has not changed, and each round the difficulty to remain consciousness will increase regardless of taking new hits or not. Even if there is no further damage, the player will be rolling for consciousness each round.
If the character retains consciousness, damage is applied normally with its normal effects.
If the character loses consciousness, Stun damage is recorded as Shock Damage and it leads to coma if the Life Point threshold is reached.
When Wound Damage reaches the Life Point Threshold, the character is
Dying, with a life span measured in minutes, so consciousness is
perhaps a lesser problem, but is still there to be dealt with if
Conditions such as injury persist until treated, or until they (Stun and Shock) naturally drop below the Life Point threshold according to the rules on page 99, and/or real medical treatment has been applied allowing for Endurance rolls of greater duration and effect.
Note: It could be argued that new injuries should also cause a roll for consciousness. As subsequent injuries will naturally occur at different points in the initiative cycle, they would also have an effect on when the consciousness check might be made. Such an argument takes the discussion into the realm of houserules and out of the scope of this question, I believe.
So to sum up:
Yes, when a character initially takes damage in excess of Life Points, a roll to remain conscious must be made, and made at that time.
Each turn that damage is in excess of Life Points will require a consciousness check, at a rising difficulty (+4)
The effect of succeeding on the roll is consciousness for 1 Turn. It will cycle through with Initiative. If the initial attack which caused the condition of Damage>Life Points occured after the character's action, then the roll will continue to follow their action. If it occured before, it will continue to precede the character's action.
If additional damage is scored, this could be argued to have an effect on the timing of the consciousness check, but this is not a feature of the rules as written. Ultimately, the player will be rolling consciousness regardless, as damage already exceeds Life Points.