Jon came across an army enlistment record for a John N. Mace, which shows that he enlisted on 26 September 1849, and deserted on 5 October of that same year. I noticed that there was a relatively high number of desertions, so perhaps there was a good reason.
This John Mace is described as being 26 years old (born around 1821), at 5'8", with brown hair, gray eyes, and a dark complexion. His birthplace is listed as Greenland, New Hampshire.
The name is very nearly right (the name on his marriage record to my 3rd great-grandmother Sophronia BLY gives his middle initial as "M"). He was the right age, and his Army enlistment/desertion coincides with the period that he "disappeared": between the birth of their youngest child, my 2nd great-grandmother Lizzie in 1847 and the marriage of Sophronia to a Jonathan WILLEY in February of 1850. The disappearance had probably occurred closer to 1850; a 32-year-old woman in the mid-19th century doesn't marry a man 40 years her senior because she wants his body. Widowed women-- and men too, for that matter-- usually had to remarry quickly; women needed financial support, and men needed someone to run the household and take care of dependent children.
So, if this was in fact my John MACE (which I think is quite likely), why did he join the army? Why did he not return to his family when he deserted? To desert was a crime for which you could be arrested (one other chap on this document actually was), so perhaps he had to make himself scarce for awhile. Maybe he intended to go back to his family, but something happened to him.
I'll probably never know the answers.
But Jon is tenaciously digging and contacting town clerks, and he's already found some more interesting potential relatives to John Mace. There is a Thomas MACE (b. 1792) who married a Catherine LIBBY (b. 1796) on 7 December 1817. They were from Greenland, New Hampshire, and would be the correct age to be John's parents. John's eldest son, who fell at the Battle of Williamsburg, was named Thomas (1843-1862).
Just leads so far, but we'll see if anything solid turns up. I want to publicly thank my new 4th cousin once removed for reaching out to me and for his detective work.