Bell X-2 and James Garner
The rocket plane at the beginning of the movie is the Bell X-2 with a dual cockpit. The original X-2 was a single seat aircraft and was the first aircraft to break mach 3 before crashing in 1956.
The X-2 was featured in the 1956 film "Toward The Unknown" starring William Holden and a very young James Garner in his first major role. Garner plays USAF test pilot Joe Craven who is killed in the crash of a Gilbert XF-120. This fictional aircraft was actually the Martin XB-51 experimental bomber. TTU was filmed mostly at Edwards Air Force Base in California. The opening scene of "Space Cowboys" is set at Edwards.
I was hoping there would be some reference to "Toward The Unknown" in "Space Cowboys" but I saw none.