I used William Frayne Amann's 1961 "Personnel of the Civil War," Vol. II, The Union Armies pub. New York: A.S. Barnes and Company, on page 154 of his List of Synonyms of Organizations in the Volunteer Service of the United States in the Missouri section which states that "Red Rovers, Hiram A. Rice's Independent Cavalry Company" later became Company I of the 10th Cavalry Missouri State Militia, which later became Company I of the 3rd Cavalry MSM (new), much as you suspected.

Bruce Nichols