I found the GriffinBibler Family Tree, http://trees.ancestry.com/tree/12082435/person/-338491927http://trees.ancestry.com/tree/12082435/person/-338491927 family tree which contains your Smith Family. This Family Tree has pictures of Otis and other family members.
In 1860 Otis's parents and siblings were living in Salem, Champaign, OH.
Cyrus Smith, b. NY, 60, physician.
Mary, wife, b. OH, 44.
Elizabeth, b. OH, 19, single.
Amelia, b. OH, 17, single.
Otis W., b. OH, 14.
Anna M., b. OH, 13,
Benjamin G., b. OH, 7.
An Otis W. Smith, 18
Born: 4 Oct 1844
Enlisted: 8 Aug 1862
Rank at Enlistment: Private
Enlistment Place: Champaign, OH
State Served: OH
Survived the War?: Yes
Service Record: Promoted to Full Corporal on 1 April 1865.
Mustered out on 14 Aug 1865, Louisville, KY.
Death Date: 10 March 1923, Yountville, CA
Source: U. S. Civil War Soldier Records & Profiles, Official Roster of Soldiers of State of Ohio, Deeds of Valor. How Our Soldier-Heros Won the Medal of Honor, Medal of Honor Recipients 1863-1994.
He married in 1874, Olive, but no location shown.
Source: Griffin/Bibler Family Tree.
Otis W. Smith was living in Terraville, Lawrence, South Dakota in 1900.
Otis W. Smith: b. OH Oct 1845, father born=NY, mother born-OH, day laborer, marr 26 years but only family member listed.
In 1910 an Otta W. Smith [how shown on index] was living in Huasna, San Luis Obispo, CA.
Otta W. Smith, b. OH, 63, farmer.
Sarah, wife, b. IL, 58, marr 48 yrs, had given birth to 10 children but only seven alive in 1910.
James Smith, b. SD, 25, single, general farmer.
Wm. O., b. SD, 18, single, general farmer.
Ruth Over?? [can't read rest of name] granddau b. SD, 6
MIllon Over??, grandson, b. CA.
Name: Otis W. Smith
Born: abt. 1843, OH
Description: 5' 6" fair complexion, blue eyes, gray hair
Read & Write? yes
Admitted: 1916, 73 yrs old
Enlisted: Aug 8, 1862, Taylortown, OH
Co & Regiment: Y, 95 Ohio Infantry
Discharged: Aug 14, 1865, Louisville, Kentucky
Cause of Discharge: S. O. 45
Disabilities When Admitted: chronic constipation, old cerebral hemorrage, left arm & leg arterial scherrois & cardiac hypentension [?]
Admitted: National Home Disabled Soldiers, Sawtelle, Los Angeles Co., CA
Remains: Sent to Sara O. Smith, widow April 6, 1923
Source: U.S. National Homes for Disabled Volunteer Soldiers 1866-1938.
The San Luis Obispo County Genealogical Society, www.slocgs.org/obit.index.htm provides copies of obituararies for $4/copy.
There is a birth date discrepancy between yours and the Civil War info shows 4 Oct 1844. His birth record in Logan County, Ohio will prove which one is correct. Successful Searching!