TriOmegaZero
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No.
Swallow Whole (Ex) wrote:
If a creature with this special attack begins its turn with an opponent grappled in its mouth (see Grab), it can attempt a new combat maneuver check (as though attempting to pin the opponent).
It must begin its turn with the creature grabbed to use Swallow Whole. This is further backed up by the following, which removes that requirement.
Fast Swallow (Ex) wrote:
The creature can use its swallow whole ability as a free action at any time during its turn, not just at the start of its turn.
So a hasted creature cannot Grab and Swallow Whole on the same turn. Only a creature with Fast Swallow can.