This is a broad topic with multiple answers to it because the South was and remains a large place with a lot of different moving parts to it. The subject can get a bit confused since white southern Unionists came in different varieties, and had different motivations.

One of the larger groups was made up of what we sometimes call "conditional Unionists." These are people who oppose leaving the Union, provided certain conditions were met in the North. You find a lot of these types of individuals in the border and middle South - places like Virginia, Tennessee, and North Carolina. Getting hard figures is difficult, but most of these people probably either fell in line, or valued their own safety enough to keep quiet. In Lexington, Virginia, home of Virginia Military Institute and Washington College - today's Washington and Lee University - only one person voted against ratifying the decision to secede in a town that had previously been known as a Unionist stronghold. Even though the first convention vote for secession actually failed in Virginia, many people who opposed secession even up to the second vote ended up falling in line with their state in the end.

Beyond this, defining who gets to be called a Unionist in the white South can be tricky in much the same way defining someone as a Loyalist during the Revolution can be tricky. The line between someone being a Unionist, neutral, or apathetic is not always so easy to discern. What we can say is that there were numerous whites behind Confederate lines who were not always enthusiastic about the Confederate war effort. Some of these people were members of German Anabaptist sects such as the Mennonites, who like the Quakers, opposed violence. Others were old-fashioned Whig nationalists whom the world had left behind. These were people who loved the Union, who remembered the nationalistic fervor of the Mexican War, and some of whom even remembered the War of 1812. Falling into this category didn't mean that somebody was necessarily likely to support the Union, but it did include people like Winfield Scott, a Virginian who was at least nominally in command of the Union army at the outbreak of the war. Others were descendants of old Federalists who clung to their social clout in little corners of South Carolina and other pockets of influence throughout the coastal South. Like some of the old Whig nationalists, these were people whom the age simply had left behind. They held to older values that simply on longer resonated with the South as it existed then. This included South Carolina's James Pettigru, who is said to have quipped that his state was too small for a republic, and to large for an insane asylum. Others were people of northern extraction who happened to be living in southern states.

There are also recorded instances of outright resistance to Confederate rule. The most famous example is the state of West Virginia, carved out of Virginia, but there were also mountainous regions of Tennessee and North Carolina that were fairly hostile to secession. There were also "Red String Bands" active in parts of the South where slavery was less important that actively favored an end to the war an reconciliation with the Union. A more dramatic example of this is Jones County Mississippi, which seceded from the Confederacy, although this was rare. Other white Unionists in the South, like Winfield Scott and George Henry Thomas, fought on the Union side, and there is evidence of white Unionists enlisting in Union regiments as well.

There were also white Unionists who packed their bags and left, including George Junkin, the president of Washington College in Lexington, who spent the war years in his home state of Pennsylvania. Not everybody would have had the means to leave quickly, however, and those who stayed behind generally had to keep quiet until Union troops arrived. I know of no large scale migrations at this time, and that is something I would have to look into. This was not necessarily everyone's favored option, however, as many people were just as tied to their own communities as they were to Unionist nationalism. Anti-secessionists in remoter areas like East Tennesee, for example, also would have had a difficult time getting into Union-held territory for quite some time. Many white Unionists still saw their state as their home, and just did not leave.

Readings

Kenneth Noe, "Red String Scare: Civil War Southwest Virginia and the Heroes of America"

William Freehling, The South vs. The South: How Anti-Confederate Southerners Shaped the Course of the Civil War

Richard Nelson Current, Lincoln's Loyalists: Union Soldiers from the Confederacy

James Broussard, The Southern Federalists: 1800 - 1816

Peter Carmichael, The Last Generation: Young Virginians in Peace, War and Reunion

Jenkins and Stauffer, The State of Jones: The Small Southern County that Seceded from the Confederacy

Daniel Crofts, Reluctant Confederates: Upper South Unionists in the Secession Crisis