I checked "The Army of the United States, Historical Sketches of Staff and Line with Portraits of the Generals-in-Chief" edited by Theophilus Rodenbough and Willaim L. Haskin. This was first published in the 1890's but was reprinted by the Argonaut Press in 1966.
On page 451, the book says the 3rd Infantry was moved to Fort Snelling beginning in May 1891. However, the Indian Wars ended in 1890.
Between 1891 and 1894 the regiment or portions of it were involved in fire fighting, the World's Fair and operations regarding the labor troubles of the 1890's.
Infantry regiments had 10 lettered companies in the 1890's. There was a reorganization in 1898 adding two more companies.
The 3rd participated in the Santiago Campaign, in Cuba in 1898.
Your ancestor's records may be found at Ancestry.com. They have recently digitized many government and non-government military records. For example for wars like the SpanAm, many states published book length lists of the citizens who particpated and their records.
Government records are at the web site below. Yhis site will guide you through the request process. In addition to rosters, pension records detail military service and may be of interest to you.
First gather whatever data you have (date of birth, death, home state, birthplace, etc.).
Good luck searching.
James B. Ronan II
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