The Henry Putnam who served in the 6th Illinois Cavalry, Company D, was the son of a Sarah Putnam. He was from Hamilton County, Illinois. He was discharged due to disability (chronic diarrhea) on 20 Aug 1863. He passed away on 29 Nov 1863.
https://www.fold3.com/image/263524144
That being said, I think you might already be conflating two different Henry Putnams. The one who died at the insane asylum appears to have been married to a Laura Putnam in Ohio, with a daughter Anna or Alice born about 1862. The other is living in Illinois and married to a Miranda (Wilcox) Putnam with a son Charles born about 1860 and a son Frank appearing later with his mother and grandfather in New York, born about 1867. Not to say he couldn't have moved and gotten remarried and weird family mixtures... but what do you have that connects the two?