The CFII who did my instrument rating was a USAF pilot on AD doing her initial C-130 training (I'm in Little Rock). It was a little unusual in that she had her CFI/CFII and other ratings, taught & flew charter before getting picked up for an ANG C-130 pilot slot. She was an exceptionally competent pilot and great teacher. But the situation is definitely the exception rather than the norm.
DOD physicians, who are of course offices (Army, USAF, USN) can sometimes moonlight for $$ in the civilian world if they're properly licensed in their state, qualified & there is opportunity. I think it's the same for nurses. They have to have command approval, they have to sign off a bunch of stuff acknowledging that their military duties take precedence over civilian work (even on short or no notice) and they have to have malpractice insurance.
So yes it's possible to work on the outside while you're an officer if you are professionally qualified, if you have your command's permission and if work is available. But it's a hassle, and there's no guarantee except military duty always comes first - even if it's a short notice or no notice change that is a bad deal for your .civ job. Seems like it would be really hard to do in flight training, if possible at at all.