A popular theme in alternate history has been the CSA falling apart within a short time of becoming independent under the principle that the individual states would be entitled to secede or because certain of those state managed to defeat 'Federal' troops from Richmond.
After which the Union swallows them one by one...salivates profusely
Oops! Sorry! Unneeded diversion!

Assume the CSA wins by wearing out Union resolve, followed by a negotiated settlement with the failed one term Lincoln administration in December 1864, rather than a decisive victory earlier.
The CSA acquires Kentucky, Oklahoma and modern day New Mexico at the talks.* I'm not considering dividing New Mexico and Arizona into northern and southern territories for a reason.** Also a certain degree of gold in damages.
Post 1864...
The CSA faces a degree of armed resistance from recently freed slaves and for years to come killings are reported. Had the Union not abandoned most of the former(and future) slaves and particularly so many of the colored soldiers this would likely not have been the case.
Both sides work on repairing their economies.
The US fields a permanent army of 40,000 while the CSA decides on 25,000. The CSA doesn't spend much time on a navy while the US retires most of her sailing fleet, instead deploying a squadron of eight ironclads*** along with @40 sailing ships with steam auxiliaries. Effectively the fleet is more modern and effective than the 1860 fleet but otherwise comparable in size except for the ironclads.
France is able to maintain Maximillian at the price of an unfortunately large number of troops. Ironically CSA interest in a weakened Maximillian selling off territories accessing the Pacific combined with Maximillian's own less than clear grip on reality and desire to be truly welcomed by the Mexican people renders the CSA willing to provide a trickle of aid to Juarez, just enough to keep him going on top of USA (reduced) aid.
No wars over Canada, no British invasions of California, no mysterious time travellers with automatic rifles, sorry.

It is 1866 which sees the unexpected, Confederate Civil War or Third American Revolution, depending on which side you're on.
The first major development is the filibustering expedition to seize Cuba for the Confederacy. Surprisingly the fleet carrying nearly 5000 men is able to get past Spanish naval patrols(rumors of a deliberate Spanish trap will persist for decades) and land successfully in Cuba. Unfortunately, shortly after landing at Bahía de Cochinos

the entire force is rounded up with extemely heavy casualties by the Spanish.
This has a major affect on the Confederate elections as partisans supporting Louis Wigfall demand that either the outgoing Davis administration force Spain to accept terms(Madrid is most unlikely to yield) or they will take action after winning the election.****
Alas the election is won, in a surprisingly close race*****, by Robert E Lee instead, who makes clear he has no intent of declaring war on Spain or making any other major effort to damage the CSA's international position. Even worse, Lee's platform included certain measures aimed at slowly phasing out slavery over an extended period.******
As a result an outcry sweeps the south, with Wigfall's supporters rallying separatists concealed as further filibustering expeditions. While historians generally agree this was only intended as a bargaining chip and Wigfall had absolutely no intention of starting a war the response by the outgoing Davis administration to call up 100,000 troops, both to field an army but also to prevent many CSA federal armories from being overrun(as several were) set the stage for a crisis and the unplanned rival shows of force which erupted into a startling degree of bloodshed outside Memphis led the governments of every state which voted for Wigfall save Arkansas to vote for 'separation from the CSA pending resolution of the current strain relationship'. If resolution could be achieved. After the seizure of two of the first CSA ironclads under construction outside New Orleans...
The US is distracted at this time by the final settlement of the Alabama affair with the British, the British are paying a higher level of damages******* because the Alabama was never actually taken by the USS Kearsarge but also because the British want the US to have enough gold as to encourage the purchase of Alaska from Russia. The idea is that since the US would not help arrangements with either London or St Petersburg if the US started meddling in CSA affairs this should keep the US out of Confederate hair for at least a year.
*The name Jim Lane, already a source of strong feelings in OTL, is held in even greater hatred(in the CSA) or esteem(elements of the USA) as his brutal evacuation of several Missouri counties arguably tipped the vote in favor of the USA.
**US refusal to allow a plebiscite in Maryland or budge at all on West Virginia almost brought down the talks, threatening the Lincoln Administration with the likelihood of the incoming Democratic president offering better terms to the CSA, hence the dramatic decision to yield New Mexico.
***The ironclads are not intended for use far from the US coastal waters. Any suggestions as to the design chosen would be welcome.
****The likelihood of the CSA waging successful war on the Spanish in Cuba in 1866 was effectively nil. The vastly superior Spanish fleet precluded any likelihood of a successful invasion with a near certainty of reprisals against Confederate shipping and port cities. Nor was there hope of the CSA building up a fleet in the next few years which could match Spain without triggering a Spanish response and potentially harmful arms race.
*****Lee carries Virginia, Kentucky, North Carolina, Tennessee, Florida and Georgia while Wigfall takes Texas, Louisiana, South Carolina, Alabama, Mississippi and Arkansas. Electoral margin was greater than the popular. Some questions about the vote count in Florida existed.
******Basically all children born to slavery a certain number of years after the law is actually passed will be freed at the age of 21 with a modest government effort to fund the liberation of older slaves and a final liberation of all remaining slaves when the last children reach age 21. In the most unlikely event the law was actually passed right after Lee took office and not overturned in the courts the earliest slavery would have ended would have been 1890.
*******Roughly 20 million dollars in gold instead of 15 would be paid.