A second use of the ability won't speed up a target's end, but it might provide a back-up in case the first use is removed. However, that's a risky strategy.
I'm at loss as of how to interpret the wording of the Sibriex ability. If it has hit a creature with Warp Flesh, can it speed up the death process by using the ability again? I'm aware that normally a creature can't be "double-poisoned". If it is poisoned and gets hit with the condition again, the duration is simply reset.
Even before you need to worry about the target being 'double-poisoned', you have to consider that Warp Creature is a "monster ability". As such, it falls under the "game features with the same name" rule from the DMG errata (emphasis mine):
Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap. For example, if
a target is ignited by a fire elemental’s Fire Form trait, the ongoing fire damage doesn’t increase if the burning target is subjected to that trait again. Game features include spells, class features, feats, racial traits, monster abilities, and magic items. See the related rule in the “Combining Magical Effects” section of chapter 10 in the Player’s Handbook.
Since this rule specifically refers to the fire elemental’s Fire Form trait, let's use the relevant part of that to model how we should approach Warp Creature.
The first time it enters a creature's space on a turn, that creature takes 5 (1d10) fire damage and catches fire; until someone takes an action to douse the fire, the creature takes 5 (1d10) fire damage at the start of each of its turns.
Consider the following case: a fire elemental moves into a creature's space for the first time. That activates the relevant portion of its Fire Form trait. Moving into the space is the trigger, and "catches fire" is the ongoing effect, "someone takes an action to douse the fire" is the end condition, and "until the end condition is met" is the duration. In addition, a consequence (sub-effect?) of the "catches fire" effect is that the creature on fire "takes 5 (1d10) fire damage at the start of each of its turns."
Now, suppose the fire elemental moves out of that space (or the creature moves on its turn, leaving the fire elemental behind), but on the elemental's next turn, no one has doused the creature and it is still on fire. The fire elemental then enters that creature's space again. What happens?
It could be that the Fire Form trait simply doesn't work on a creature already under its effects. However, the rules do NOT say that the trait cannot be used; rather, they say that of the two instances of the trait, the effects of only one of them apply while the durations overlap. Thus we see that even though the creature is now being affected by two instances of the Fire Form trait, it can have only one "catches fire" effect active.
Now let's apply this to Warp Creature. Here we have an ability (Action/Legendary Action) rather than a trait, but the rule specifically covers this case as well.
The sibriex using the action (or legendary action) is the trigger, there is an initial save which grants immunity if made, but if failed, "(this sibriex) poisoned" is the ongoing effect. '(This sibriex) poisoned' is like the poisoned condition, but in addition has the consequence (sub-effect) of requiring a save each turn, and then those saves produce other (sub-sub) effects. The end condition of the "(this sibriex) poisoned" effect is "three successful saves against the poison", and the duration is "until one of the end conditions is met".
So, can a sibriex use Warp Creature on a target already under its effects? Yes, it clearly can. This is an additional use of the same ability, permitted like the additional use of the Fire Form trait on the same target.
What happens if it does? First, the target gets an initial save for the new ability. Note that if this initial save is made, the target "becomes immune to this sibriex's Warp Creature". This is a risky strategy for the sibriex. Whatever benefits it might get from another instance of Warp Creature on the same target will need to be balanced against the chance of giving the target immunity to not only the new ability, but the one it is currently under. In seeking to enhance its ability, it might end it prematurely.
Next, If the target fails the save, it is then under the influence of 'two or more game features with the same name while the durations of the effects overlap.' Would this failed save give it an additional level of exhaustion? Certainly not from the new ability, because the poisoning and exhaustion are effects, and only the active (first use) of the ability can currently affect it. But could the active effect count the failed initial save of the second Warp Creature as a 'failed save' that triggers a level of exhaustion?
Three successful saves against the poison end it, and ending the poison removes any levels of exhaustion caused by it. Each failed save causes the target to suffer another level of exhaustion.
Obviously "each failed save" does not include saves against a different spell or ability. It is referring to the preceding sentence, 'saves against the poison'. The initial save for the second Warp Creature ability is not a save against the poison effect, it is a save against the Warp Creature ability, and the poison effect does not trigger until it fails this save.
No help there to a sibriex that wants to use its ability a second time to speed up the demise of a creature.
What about insurance against a creature making the three poison saves?
Three successful saves against the poison end it, and ending the poison removes any levels of exhaustion caused by it.
What does ending "the poison" mean? Does it mean ending the '(this sibriex) poisoned' effect from only the active Warp Creature ability, or does it mean ending as well the potentially poisoned effect from the Warp Creature ability that is waiting to take effect when the first one ends?
Also, consider that the target or its allies might, at some point, use an ability to end the poisoned condition, especially since this would end the resultant exhaustion. Would this end just the active effects, or the ones from the inactive Warp Creature as well?
Unfortunately, this is an unresolved question in RAW at present. Does a feature that ends a condition, end only the active condition, or all the same-named simultaneous inactive conditions?
Since it is unresolved, it would be up to the DM to determine. If the DM ruled that anything ending the active Warp Creature did not affect any inactive ones, then certainly a Sibriex might derive some benefit from multiple uses of Warp Creature, so as to have one 'waiting in the wings' when the first one was removed. Consider, however, that the sibriex gets three Legendary Actions wherein it can use Warp Creature at the end of another creature's turn. Since there is always the risk of ending the active ability, it would be safer to wait until the active ability had been removed, and then apply it anew at the end of the turn of the creature that had removed the active effects. The exception would be in the case of enemies that are actively denying it the "creatures it can see within 120 feet of it" condition it needs to use the ability. Faced with opponents making flyby attacks or becoming invisible, the sibriex might want to apply a back-up Warp Creature if the risk of not being able to target was greater than the risk of the target making the save.
If, on the other hand, the DM rules that a third successful save or some remove poison ability removes both active and inactive Warp Creature abilities, then there is no incentive at all for a sibriex to use the ability again on any creature already afflicted.