It depends on whether you want a sustainable 'Confederacy' or not
So you probably need at least the minimal 11 states that seceded. Here are a few more common and less common possibilities from 1861, I think some of these may have already been mentioned:
December 1860: Instead of resignation, Secretary of War John Floyd stays at his posts and completes the transfer of heavy cannonry from Northern arsenals to Southern installations in Galveston and Mississippi. He successfully defends the charges of fraud and conspiracy against him in March 1861 and is not available as general for the Battle of Carnifex Ferry. If the Confederacy wins the battle and might be able to push McClellan out of West Virginia altogether. Floyd might also be able to do more damage before he leaves the Union as well. Montgomery Meigs may not be able to be recalled in time to serve as quartermaster general for the Union, perhaps he is discharged outright or given a field command instead.
February 1861a: Robert Toombs of Georgia is elected President of the new Confederacy, he negotiates an end to the Fort Sumter crisis and begins to negotiate for release of the other states. Quietly he sends agents abroad and garners recognition from the UK and France before shots are fired in hostility, leading to an arbitration by the UK favorable to the Confederacy in early 1862.
February 1861b: Alexander Stephens of Georgia is elected President of the Confederacy and uses his friendship with Lincoln and others from his Congressional days to similar effect.
March 1861a: Joseph Johnston remains as Quartermaster General long enough for Robert E. Lee to be named head of the Army of Virginia and be the overall field commander at First Bull Run.
March 1861b: Nathaniel Lyon tries to use his connections in Washington DC to be named head of the Federal arsenal in St Louis, upon his attempt Lyon's superior is notified and has Lyon demoted for insubordination and transferred to Virginia. Anyone with Southern sympathies who takes the post (Perhaps this is where Joseph Johnston is stationed if he remains Quartermaster?) will likely surrender it to the pro-southern forces under Governor Jackson and avoid the Camp Jackson affair, giving the Confederacy control of the state.
April 1861: Benjamin Butler's army massacres rioters in Baltimore and sacks businesses for food. Guerillas destroy rail lines and down telegraph lines between Philadelphia and Baltimore as Butler deals with a riot that engulfs not only the police but the 6th Massachusetts militia as well. Many in the crowd become armed after a firefight later attributed to a Massachusetts militiamen causes rioting to intensify the next day. Butler occupies Federal Hill and begins using artillery to clear the streets, hoping the fear of overwhelming force will restore order in the city. This backfires, and the Maryland State Legislature approves a secession ordinance, isolating Washington DC and having Lord Lyons inform his superiors.
June 1861: Instead of boycotting the 1860 special election polls, Kentuckians turn out in droves, putting equal numbers of Confederate and Union sympathizers in Congress. Magoffin adheres strictly to neutrality and takes Kentucky out of the fight as much as possible. When 'Bull' Nelson builds pro-union Camp Dick Robinson in Garrard County, the state legislature takes this as an invasion and sign that the Union will not respect the state's neutrality. Kentucky secedes from the Union by very narrow margins shortly thereafter.
November 1861: Trent affair goes badly, UK intercedes as France throws in to announce its support of the Confederacy (its troops already gearing up for intervention in Mexico).