In the Pacific Ocean, 37°05'″N , 156°47'W 1000 miles North East of the Hawaiian island of Kauaʻi.
The halfway point between Lafayette, Georgia and Clinton, Missouri, falls in Charleston, Missouri.
The halfway point between Mt. Pleasant, Iowa, and Branson, Missouri, is about halfway between Fulton and Jefferson City, Missouri.
Dawsonville, Missouri is your halfway point.
The halfway point between Monticello, IA and Springfield, MO falls in or around Perry, Missouri.
Cave, Missouri is halfway between Fayetteville & Chicago
Kansas City, Missouri.
The halfway point between Alexandria, LA and Joplin, Missouri is Nutts, Arkansas (yes, really).
The halfway point between Clinton Iowa and Atlanta Georgia falls around County Road 1200 North, Metropolis, IL 62960, USA.
The halfway point between Chicago Illinois and Winnsboro Texas is Portageville, Missouri.
Washington DC
Riverpark, Missouri.
The halfway point is basically the area where Nebraska, Colorado and Wyoming meet. This is not an exact answer, but in relative terms this is the halfway point between Salmon, Idaho and Kansas City, Missouri. If your looking for a town, then I would say the capitol city of Wyoming (Cheyenne) is pretty much as close as your going to get to a city halfway between Salmon and K.C.