Wallace (Missoula) was granted a post office in 1892 [sic] with Elias Bryan as postmaster. Postal records say the name was changed to Clinton in 1892 but actually, the postal business for this area was merely transferred to Clinton, which is nearby.Based on information contained in Montana Post Offices and Postmasters by Dennis J. Lutz, and on the usps.gov website, I believe Cheney meant for that first date to be 1883.
I don't know what Cheney's source is for the fact that the change from Wallace to Clinton in 1892 was more than a name change, but it makes sense. The town of Wallace was established by 1883 in Wallace Gulch (area circled in red), close to the mines, while Clinton was set up as a railroad town along the east side of the tracks (area circled in blue). As the mail went from being carried my stagecoach to being carried by rail, it made more sense to have post offices located close to the tracks.
P.S. Did you know that Clinton is the most common name for a post office in the United States? There are 26 post offices with the name Clinton. The second-most-common name for a post office is Madison, with 25.