To come at this from a slightly different angle:

Did the Civil War significantly drive westward migration after the war? I'm thinking freed slaves, Confederate veterans, or people who got uprooted by the fighting.

There were definitely refugees during the American Revolution. New York City was occupied by the British for most of the war, and hundreds of Patriot families left for upstate New York and elsewhere for the duration, only returning home at war's end. Future Chief Justice of the Supreme Court Jon Jay was among these refugees and so were his parents.

Here's a list of known refugee families of New York City first published in 1976 for the bicentennial.

A few other cities underwent lengthy sieges during the war, and also produced some refugees, though the occupation of New York was the longest. The city was the base of British operations for the duration of the war.

After the war, you could make a case that at least some, if not all, of the Loyalists (i.e. Americans who fought on the British side) who left America were war refugees. A lot of them were forced off their land and went to Canada for the offer of free land by the British government because they had nowhere else to go. There were 40,000+ Americans who left for Canada. Many did this voluntarily for the opportunity but many others did it out of lack of other options.

These ex-Americans mostly settled in future Ontario and the Maritime provinces. In fact, this is the beginning of English speaking Ontario. Up until the 1780s, it was part of Quebec and mostly populated by Indians and a handful of French farmers and fur trappers.

Some of these Loyalists eventually returned back to the States when famines and disease became too much to bear. But some were not welcome back. After the first decade or so, conditions had improved considerably with the building of new infrastructure (mills, stores, schools) by the earliest pioneers and Canada began to attract more than just war refugees.