Here’s what I would suggest. Get a free account on familysearch.org.
When I’m working on soldiers, I like to just put in their name and then filter by record collection for collections that apply to the civil war. I don’t like to use place of birth, and a year range when searching military records because it often filters out some war records that you might like to see.
The Civil War soldiers index is helpful because it helps narrow down regiment or regiments that a soldier served in and sometimes it offers an alternate spelling on a name or, in extreme cases, an alternate name
For union soldiers, filtering down to and finding the 1890 military census collection helps establish where a soldier was living in 1890. If they’ve passed on and their wife is listed instead and that can be helpful, too.
Then I like to filter to the pension records. They can give clues to geography, spouses, sometimes parents or children, place of death and date of death.
Finally, if you know the state or states, the soldier has lived in, filtering to state related Civil War, or veterans records sometimes yields interesting results.
All of these military details help make sure that when you’re looking at the regular US census records, you have the right individual.