Joe,

It was a war designation and not postwar. VET stood for Veteran and the regiment you referenced must have been a Veteran Volunteer regiment.

Under the terms of the Veteran Volunteer Act of 1864, Union soldiers who decided to reenlist (for potentially another three years) received a thirty day furlough, free transportation home, a pay-off of any bounties owed to them and an additional $400 bounty. If a majority of the regiment reenlisted, it would remain in existence and then be designated as a "Veteran Volunteer" regiment.

Mike