When did the military start using SSB? I haven't seen anything military with SSB earlier than the mid to late 1950s: interestingly, that early stuff generally had both sidebands available, although some of that was to allow ISB working.
In the US, the military didn't start with SSB until the mid 50's. USAF General Curtis LeMay "Bomb 'em back to the stone age", was an active ham and asked Art Collins of Collins Radio to collaborate with SSB experiments.
They fitted up a USAF plane with both AM and USB radios and made and extensive (if not world circling) tour comparing the two modes on both military and ham frequencies. SSB won hands down. This was widely covered in ham magazines of the time. The rest is history.