Damon, since no one else has replied, I will offer what I know, assuming this is the same Union Volunteer unit.
My ggfather David D Franklin's tombstone reads"Co B 3 US VI" I have a copy of his enlistment form signed October 13, 1864 while he was in the Union prison at Rock Island, Ill. He was sent to Ft Kearney, Nebraska Territory where he apparently only did post farm duty. He had been captured at Benton, Tenn as a private in Co. C, 3 (Lilliard's) Tennessee Mounted Infantry.

Regarding your question as to who was authorized to raise this US Vol Inf at Rock Island , the following information is from the very good book"The Galvanized Yankees" by Dee Brown. Since the 1864 politics were very tense, particularly in Pennsylvania, President Lincoln authorized a Col. Huidekoper and Judge Pettis to muster Rock Island prisoners into US service after examination and taking the oath of allegiance. However, General Grant opposed this, and suggested that any such enrollees be sent to the west. Lincoln then sent Captain Henry R. Rathbone to Rock Island and later issued a memorandum allowing the examination and mustering to proceed under Col. Andrew J. Johnson (commanding ar RI) and Capt Rathbone under the discretion of Col. Andrew P. Caraher (later commander of 2nd US Volunteers). (Capt. Rathbone was later one of Lincoln's protectors when he was killed at Ford's theater). So it appears that Lincoln himself initiated the authorization.

Side comment: GGGrandfather Franklin's widow was denied a US pension because he had not been directly involved in fighting against the South.