You got a bit carried away in your first paragraph. The Red, Ouachita, Yazoo, and Atchafalaya and their tributaries were all under Confederate control until the summer of 1863.
Conversely, the White was largely under Union control as early as the summer of 1862. On several occasions Union flotillas pushed up the White. One such expedition led to the well-known fight at St. Charles in June 1862, but there were others as well, notably after the capture of Arkansas Post in January 1863.
The Confederates managed to retain "control" of the lower Arkansas even after the loss of Arkansas Post primarily because (1) no Union army or navy commander wanted to risk stirring up a hornet's nest in Washington by launching another "wild goose chase" into the trans-Mississippi before the opening of the Mississippi River, (2) the Arkansas was a seasonal river (i.e., a snowmelt stream) which pretty much ran dry in the late summer, fall, and early winter months and that made navigation below Little Rock very risky, and (3) as much as it pains me to say this, Little Rock was of very little strategic importance.
All is explained in William Shea and Terence Winschel VICKSBURG IS THE KEY: THE STRUGGLE FOR THE MISSISSIPPI RIVER (University of Nebraska Press, 2003), which would make a perfect stocking stuffer for yourself or your favorite ACW enthusiast! (Sorry about the shameless plug but I can't help myself! Somebody stop me!)