Actually, you'd be wrong. The various Southern states started rebelling and seceding almost from the day after Lincoln was elected. Between December 1860 and April 1861, numerous other US forts and arsenals across the South were taken by rebel forces. Fort Sumner SC, was manned when Union troops fell back from Castle Pinckney and Fort Moultrie (both near Sumner) and taken by Southern rebels.
Fort Pickens at Pensacola Florida had been under siege by rebel forces for almost three months when the shooting at Sumner started. The only reason it hadn't been taken by force was Stephen Mallory, who would become the Confederate Secretary of the Navy, negotiated a truce that held.
It was the dithering non-response of Buchanan that allowed all this to happen between Lincoln's election and his inauguration into office.