Great story Stix.
The sidearm your father was issued was the long-barreled U.S. Colt Model 1892 Military Double Action Revolver in .38 Long Colt.
This was the first double action pistol ever adopted by the U.S. Army. (Double action means, that the revolver will always fire by just pulling the trigger, without the need to first cock the hammer back by hand, which is what a single action revolver is.)
The adoption of this revolver by the Army was fueled by a driving desire to update its standard sidearm with something "truly modern", using a smaller higher velocity bullet and a rapid rate of fire (as the Europeans were doing.) The thought was that a higher velocity, narrower bullet would always translate into more energy being delivered to the target and better penetration, and thus better stopping power.
Unfortunately, they found out that paper ballistics and calculations almost never correspond to real life.
In both the Spanish American War and the Philippines insurrection that followed, officers found that these .38 rounds would not "stop" a charging determined assailant, even after delivering three or more rounds on target.
So disgusted were the officers by this, that many older members dug out and carried their old single-action 1873 Colt .45 Calvary Pistols, which though they needed to be cocked by hand, delivered a devastating and game-stopping .45 slug.
In 1911, the U.S. Army admitted its mistake, and switched to the 1911 Colt Automatic, designed by John M. Browning, which is still a highly desired and widely carried sidearm today.
The 1892 Colts were not thrown away, but put in storage. So, I imagine that when the U.S. Post Office employees on railroad runs were in need of sidearms, these older revolvers were issued to or acquire by them.
Often, when the Army declared a firearm obsolete, they would be sold off as surplus for as little as $5.00 each. So, I would imagine that if any railroads desired to equip their trainmen or "agents" with a sidearm, these would have been an economically attractive option.
Mannyrock
