... the local elections that week [rb note: late March or early April, I think] went badly, as Republican candidates lost in Connecticut, Ohio, Brooklyn, Rhode Island, and St. Louis. "Thirty days more of 'Peace Policy' at Washington," one Ohioan warned Lincoln, "and not only the Republican Party, but the Government itself will be gone to destruction." ...
The mounting pressure to act dismayed Lincoln. He had struggled to keep the door open for Seward's policy of delay as long as possible. The Pickens expedition was part of that effort, its secrecy designed to prevent the Confederates from getting wind of the mission and attacking the fort before the relief force arrived. To that end, the president had dissembled with Welles, feigning ignorance of orders in that simple, straightforward manner he had perfected to an art form.
That's the important point.
Lincoln's policy then was to delay and delay as long as possible, in hopes (futile as it turned out) that Southern Unionists would exert pressure on Confederates to rejoin the Union.
In the mean time, he intended to hold onto those few Federal properties still under Union control -- Sumter and Pickens.
But there was no way to hold Sumter beyond April 15 without resupplying it, and so that's what Lincoln attempted, and notified South Carolina Governor Pickens on, April 6.