By 1890, the main line of the Santa Fe railroad was complete, from Chicago IL, all the way to Los Angeles CA. This railroad passed through both Gallup NM and Winslow AZ. From Chicago, the Baltimore & Ohio and the Pennsylvania Railroad both had direct connections between Chicago and Washington DC.
The only problem is the stop over in Terre Haute Indiana. That town was located on the mainline of the Pennsylvania railroad's line between Pittsburgh PA and St. Louis MO. The Santa Fe did not serve St Louis MO. Anybody going to Terre Haute Indiana from the West almost certainly went through St. Louis. To get to the far west from St. Louis, there were two alternatives : Take the St, Louis Southwestern to Corsica Texas and change trains to the Southern Pacific railroad, or take the Missouri Pacific and its Texas Pacific subsidiary to El Paso Texas, then take the Southern Pacific to California.
The Santa Fe passes through Kansas City. There are several ways to get between Kansas City and St. Louis. They could have gone on the Missouri Pacific, the Chicago & Alton, the Burlington, the Wabash, the Rock Island or the Missouri Kansas & Texas (Katy) railroads.
Based to the popularity of passenger train travel, the Santa Fe out of New Mexico and/or northern Arizona is the most likely route chosen. Anybody that stops in Terre Haute on a trip to Washington DC almost certainly traveled on the Pennsylvania railroad.
Source: "The Classic Western American Railroad Routes" edited by Bruce Cooper
"The Cental Pacific and the Southern Pacific Railroad" by Lucius Beebe.