Up until the closure of the West, there was a significant wave of western migration after each of America's wars - the French and Indian War, the Revolutionary War, the War of 1812, the Mexican War, and the Civil War - I've often wondered if some of that wasn't a degree of self medication as well, but I don't know how you could document it.
A Danish officer/author called Dinesen* served as a LT in the Danish army in 1864 when he was 18.
Including being a platoon commander during the main fight At Dybbøl. (he wrote a book about his experience, just as his father had done about his service in 1848-50, and in north Africa with the French)
He then in 1870 traveled to France and joined them against the North German federation.
He served on a staff, but this did included messenger duty to units in combat.
The force he was part of was pushed into Switzerland where he gave his word that he would not escape
and rejoined the fighting.
So he "escaped" but went to Paris to write about what was happening. So he documented the Paris commune and watched how it was suppressed... in a horrible bloodshed, he called the worst he ever saw. (Published a book about it in 1873)
After the Franco-Prussian war he moved first to Nebraska but ended up in Wisconsin, and spend most of 73 and 74 living by himself, but did have a close relationship with a local tribe. (and he wrote about it)
He wrote that he had a sickness in the soul, from taking part in the two wars, being a spectator to the civil war in Paris.. and he just gave up and traveled to America.
In late 1874 he returned to Denmark and took over his fathers estate.
In 1885 he hanged himself.
So this is one example of just trying to get away from everything... because of a sickness in the soul.
Iam sure plenty of veterans tried the same after the civil war. (just like it unfortunately happens to day)
But yes, It would be a hard and every labor intensive job to actually document if this was common.
One would need to prove that a veteran was more likely to move west than a none veteran.
Everything else being equal.
*(the father of author Karen Blixen and Victoria Cross winner Thomas Dinesen)