It's not surprising you found that pin way out there! SAC was a very large command, and it had a global presence. After the Cold War in the early 1990s, Strategic Air Command (SAC) was changed simply to US Strategic Command (USSTRATCOM). Based at Offutt AFB, Bellevue, NE, SAC's main role was control of the nuclear triad: The Nuclear Missiles, Bombers and Submarines. (STRATCOM continues this mission with some notable differences). During the day, most members of SAC would tell you that it was rough duty--very very stringent requirements and policies; no room for error and everyone accountable for their actions. I can attest, it was rough duty. Today, old members of SAC who work at STRATCOM either still on active duty, or retired and working as a government civilian, hearken back to the those days when discipline and allegiance to the rules seemed to mean more than they do today--they were respectable times. I wore that pin a good many years until we became STRATCOM. Nice find!
PS, SAC was mostly an Air Force command, but today STRATCOM is comprised of many branches of service, foreign and domestic. From my experience, the pin was worn on the uniform, not the hat, but that could be different for different branches of service. Your badge appears to be an older version which had a pin on the back at one time. In later production, that badge would have held two post pins with "frogs" to secure it.
