The move would be its only action that turn
The Rule of Staggering Strike in the CA says (emphasis mine):
If you deal damage with a melee sneak attack, you can also deliver a wound that limits your foe's mobility. For 1 round (or until the target is the beneficiary of a DC 15 Heal check or any magical healing that restores at least 1 hit point), your target is treated as if it were staggered, even if its nonlethal damage doesn't exactly equal its current hit points. A target can resist this effect by making a successful Fortitude save (DC equal to damage dealt). Multiple staggering strikes on the same creature do not stack. This feat has no effect on creatures no subject to sneak attack damage.
RAW this is read that from the moment the strike hits the target is considered staggered providing he fails his save. This means that at the point where the next action is started after the move the target of the strike is staggered.
The Staggered conditions reads:
A staggered character may take a single move action or standard action each round (but not both, nor can she take full-round actions).
So if the target wanted to do an action after having moved it could not as at that point is was staggered and already did an move action this turn.
I would argue that if he already did an attack he could still finish his move but that is my opinion/judgement as a DM and your DM might rule differently.