Waco is a a good halfway point for that trip.
No, Ft. Hood is nearest to Kileen, Texas, about half way between Austin and Waco.
Tulsa, Oklahoma
The halfway point between Fort Worth, Texas and Augusta, Georgia falls in or around Bienville National Forest, in Lake, Mississippi.
A long way
Half way point between. Fort Hood and Fort Benning
1333 miles one way !
Depends on the way you go.
Austin
There is a daily train each way between Austin and Dallas. The southbound train leaves Dallas at 11:50 am and arrives in Austin at 6:30 pm. The northbound train leaves Austin at 9:31 am and arrives in Dallas at 3:20. If your destination is also close to Fort Worth, you can shave over an hour off of this travel time.
Love Field serves Dallas, but for long haul flights, planes fly into the Dallas-Fort Worth Airport, which is roughly half way between those two cities.
Kingsbury,TX. located about half way between SanAntonio,TX. and Austin,TX.
It appears that Fort Myers is the approximate half way mark.