There are 2 points worth making.
First, the Federals seem to have had a lot of men east of the Mississippi who were available to parry any offensive by the Conferates in the trans-Mississippi. Obviously one example is the portion of the XVI Corps under A.J. Smith. He had been fighting in Mississippi in a successful attempt to tie down Forrest and prevent him from cutting Sherman's supply lines. But after Smith's men moved north to Missouri, additional Union forces from Louisiana took Smith's place to assist Steele.
Second, even though the Federals were blessed with steamboats and railroads, Union screwups could still thwart slow down the arrival of reinforcements. Initially A.J. Smith reinforced with troops under Mower Steele to keep the White River open. When it became clear that Price was to the north and was gathering his forces, Steele made a bad mistake. Instead of putting Mower on steam boats, he sent him after Price.
Mower spent the next three weeks fruitlessly chasing Price through the swamps of northern Arkansas. Even worse, the elite cavalry under Winslow were stuck in the swamps too and were not available for use against Price until the third week in October. Pleasonton and Rosecrans were deperately short of first-rate cavalry and could have used them to stop Price in southeast Missouri.