Dear Ms. Alex,
Thank you for posting your request on History Hub!
We searched the National Archives Catalog and located three series of the 1st Infantry Regiment: Orders and Special Orders Issued by the Department of Texas, 1856-1860, Letters, Orders, and Reports Received from or Relating to Enlisted Men and Officers, 1861-1872, and Copies of Letters Sent, 1847-1870 in the Records of U.S. Regular Army Mobile Units (Record Group 391) that may have information about his move to Kansas from Texas.
We also located two series the Enlistment Papers, 1798 - 10/31/1912 and Registers of Enlistments in the United States Army, 1798 - 1914 in the Records of the Adjutant General's Office, 1762 - 1917 (Record Group 94) that should list your great-grandfather.
We suggest that you review the U.S. Census Bureau’s 1860 Overview - History; and 1860 Instructions - History website for background information about this census. It may be possible that your ancestors were traveling from Texas to Kansas on June 1, 1860 when this decennial census was conducted as you have not been able to locate a census record for them.
You may wish to also review The Regular Army Before the Civil War, 1845-1860; and Infantry, Part I: Regular Army / The Germinal Period, 1816-1860 websites for supplemental information about the U.S. Regular Army and the 1st Infantry Regiment.
We hope this is helpful. Best of luck with your family research!