I have researched historic forest fires in New Mexico from old newspaper articles, roughly from the Rio Grande near Socorro, NM west to the AZ line, including Springerville, AZ and the White Mountains. This is mostly today's Cibola and Sitgreaves National Forests. There were several HUGE fires in that area in the 1880s and early 1900s that burned 100,000s of acres of forest. There just was no fire suppression in those days. For the handful of people living in the small towns in the region effected, such a huge fire was just too daunting to tackle. Most were reported as lightning caused. So, they burned for weeks, even months, until the monsoon rains arrived to extinguish the flames. Forest Service administration began ca. 1910s, though mostly involved with surveying assets and managing homestead claims. In the 1920s, the Forest Service expanded to include land and timber management, and some fire protection. For example, in 1924, the Magdalena (NM) district headquarters for a million acres of forest only had seven Forest Service employees and 20 part time summer workers to work the forest, including addressing any fires. Fire lookout towers were built at this time with the hope fires could be detected and extinguished before they grew beyond a few tens of acres.
There are a few instances of fires along the Rio Grande that threatened the then mainline track of the AT&SF (SG). The railroad, indeed, sent all available employees and recruited local men to tackle the fires, which were usually extinguished and the track reopened in a day or two. In the forests and timberlands with no railroad, this was not a luxury. The newspaper articles on these fires are fairly brief and lacking detail, except where a lumber mill or something was destroyed, suggesting such fires were just taken as an accepted occurrence in those days and scarcely worth reporting. I did find one article that stated the railroad (AT&SF in this case) had agreements with many of the towns they serviced that they could use water from the AT&SF water tanks to extinguish a town fire (as water generally scarce) in exchange for labor should the railroad need their help.
I suspect most fires before the 1920-30s, where there were no railroads, were like the above fires - there was no choice except let them burn until Mother Nature extinguished them with the Autumn rains. As a result, little documentation exists to preserve the history of such fires.