It didn't help that, when I wasn't a foot soldier in Hayes's Army, I
was still serving in the Reserves.
The comma-separated phrase "when ... Army" is a "parenthetical phrase" -- it adds information which, while perhaps necessary for comprehension, is not needed to have a "correct" sentence.
So analyze "It didn't help that I was still serving in the Reserves". "It didn't help" is the syntactic "meat" of the sentence, and "that ... Reserves" is a conjunctive clause (may be called other things, depending on your religion) that identifies what "It" means.
So "It" is the subject "did not help" is the verb, and "I was still serving in the Reserves" tells us what "It" means.
"Reserves", especially when capitalized, would be taken to mean "US Army Reserves", people who work "normal" jobs but who can be "called up" on short notice to serve in the military. They generally must attend training sessions at regular intervals, somewhat disrupting their "normal" lives.
It's difficult to say with any certainty but I would interpret "Hayes's Army" to mean that "Hayes" is a business manager who expects military-like obedience and subservience from his underlings. "Foot soldier" implies that our narrator was in the lower ranks of Hayes's organization.