There was a 10th Regiment of Ohio Cavalry, but I don't see him connected with that unit.
There was a Solomon Wyatt with the 11th Missouri Cavalry (Union) who went in and came out as a Private in Company I.
11th Regiment, Missouri Cavalry
Organized at Benton Barracks and St. Joseph, Mo., March 28 to December 11, 1863. Attached to District of St. Louis, Mo., Dept. of Missouri, to December, 1863. District of Southwest Missouri, Dept. of Missouri, to January, 1864. District of Northeast Arkansas, 7th Army Corps, Dept. of Arkansas, to May, 1864. 3rd Brigade, 2nd Division, 7th Army Corps, to September, 1864. 3rd Brigade, Cavalry Division, 7th Army Corps, to February, 1865. 2nd Brigade, Cavalry Division, 7th Corps, to March, 1865. Separate Cavalry Brigade, Cavalry Division, 7th Army Corps, to July, 1865.
SERVICE.-Duty in District of St. Louis, Mo., till December, 1863. At Springfield and Rolla, Mo., till February, 1864. Expedition from Springfield to Huntsville, Carrollton and Berryville, and skirmish, November 10-18, 1863 (Detachment). Operations in Northeast Arkansas January 1-30, 1864. Martin's Creek January 7. Rolling Prairie January 23 (Co. "B"). At Batesville, Ark., February to April. Expedition from Batesville to Searcy Landing January 30-February 3. Morgan's Mill, Spring River, White County, February 9 (Detachment). Independence, Mo., February 19. Waugh's Farm, near Batesville, February 19. Expedition from Rolla to Batesville, Ark., February 29-March 13. Scout from Batesville to West Point, Grand Glaze and Searcy Landing March 15-21 (Detachment). Expedition from Batesville to Coon Creek, Devil's Fork, Red River, March 24-31. Van Buren County March 25. Scout from Batesville to Fairview March 25-26 (Detachment). Near Cross Roads March 27. Spring River, near Smithville, April 13 (Detachment). Jacksonport April 20. Expedition from Jacksonport to Augusta April 23-24. Near Jackson port April 24. Ordered to Duvall's Bluff May, 1864, and duty there till October. Scout in Craighead and Lawrence Counties June 25-26 (Co. "M"). Clarendon, St. Charles, June 25-26. Clarendon June 27-29. Scout to Searcy and West Point July 26-28 (Detachment). Des Arc July 26 (Detachment). West Point July 28 (Detachment). Hay Station No. 3 July 30 (Detachment). West Point August 5. Expedition from Little Rock to Little Red River August 6-16. Operations in Central Arkansas, with skirmishes August 9-15. Duvall's Bluff August 21 and 24. Long Prairie August 24. Jones' Hay Station August 24. Duvall's Bluff September 6. Searcy September 13. Expedition from Duvall's Bluff toward Clarendon October 16-17 (Detachment). Brownsville October 30. Duty at Brownsville till February, 1865. Expedition from Brownsville to Augusta January 4-27, 1865 (Detachment). Moved to Little Rock February 4, and duty there till June. Moved to New Orleans, La., June 27-July 3. At Greenville till July 27. Mustered out at Greenville July 27 and discharged at St. Louis August 10, 1865.
Regiment lost during service 2 Officers and 28 Enlisted men killed and mortally wounded and 5 Officers and 181 Enlisted men by disease. Total 216.
Also in this unit:
Joseph W. Wyatt, Co F, Cpl, Sgt
Parnell K. Wyatt, Co F & C, Pvt, Pvt
There was also a Solomon Wyatt who served in the 48th USCT, Co C.