Jackson is in New Orleans threatened by a British army. He's the executive on the spot, and he feels compelled to suspend the Writ.
If the victory at New Orleans was a close run thing and the suspension of the Writ was one of the items vital to that victory, should Jackson have just resigned himself to defeat because there was no way he could get the Congress to suspend the writ in a timely fashion?
You are relatively new here, so you may not have seen the oft-posted assertion by the current Chief Justice that presidential power to suspend the Writ has still yet to be definitively answered.
Walt
See my post #485.