Worn N/S track rod end.

I would suggest worn suspension bushes on one side allowing the suspension to move when torque is applied, probably on the opposide side to which it pulls when you accelerate as the wheel will move forward. If you lever the suspension fore and aft with a crow bar you should be able to see it move.

This effect of going from under power to overrun producing a sideways effect does sound like something allowing shift in the tracking, unlikely though that is.

You could try a simple check which shows any wear problems quickly, simply park the car and taking each wheel in turn try to move the car by firmly rolling each stationary wheel forwards and back, any play or wear in the suspension will show itself by the wheel that is at the problem end moving considerably more than the others inside the wheelarch, then investigate further if you find extra movement.

Interesting that swapping the tyres improved on the constant left side pull though, i would have thought it was still on original tyres, has the OP checked that the tyres are indeed the usual make and model fitted to other Santa Fe's, and are any of the tyres showing unusual wear.