In this scenario, let's assume that the exact line of events around the time of Fort Sumter being attacked means TN manages a bare majority against secession in the second vote. This is not a minor change.
In the first vote on secession before Fort Sumter, Eastern TN was against, Western TN was in favor, and Central TN was barely against. When Lincoln called for volunteers after Fort Sumter, a second vote was held and Central TN overwhelmingly voted for secession.
In this scenario, for whatever reason Central TN stays pro-union (maybe the fire eaters did something very beyond the pale, or Andrew Johnson was still governor). I see men in Eastern TN and some in Central TN volunteering. I also see Western TN men leaving the state to enroll in Confederate units. There is probably also an attempt by armed Confederate sympthizers to take over the state entirely. At minimum, a a Confederate government of TN will form, claiming to be the leigitmate state government.
At that point, Federal troops will need to be sent into TN to keep it secure. IOTL, TN seceded in early June. Bull Run was in July 21. Grant did not move down tthe Mississippi until November with his battles at Fort Donelson until February 1862. ITL, someone needs to be sent to TN ASAP. Union troops might leave southern IL sometime in June 1861 (perhaps commanded by Grant). Kentucky is simply in the way, so how the Federal Army and Lincoln treats it is going to be important, but it's possible that Grant is able to traverse the Missippi River and secure supply lines without driving Kentucky into the Confederacy. The Kentucky legislature is definitely pro-Union.
With Union troops in TN, it is now possible for the Federal Army to recruit volunteers. Previously, there may have been volunteers ready to join, but no actual way for them to be armed and trained.
At this point, the next stage is when Confederate Armies invade. It's likely this will happen by September to November 1861. Let's assume Grant wins. So either in late 1861 or early 1862, the Union will have several goals. March south on the Mississippi River to cut the Confederates in two, advance into northern Alabama to recruit Unionists there, or possiblly try to march on Atlanta. The Mississippi River is the primary goal at this stage.
By the end of 1862, the Union probably controls the entire Mississippi River and has opened up northern Alabama for recruiting. Atlanta is threatened and will likely be the target of the 1863 campaign.
The Confederates will be in a bind because the losses here will drain strength away from Lee in an attempt to hold back Grant and other Union forces. The war might be shortened by a year, but there are many unknowns that can happen.
I don't know a lot of the specifics in this area or time, but I am sure there are many people who can elaborate more and point out mistakes above or opportunities that I missed.