You may not moonlight/perform off-duty employment (ODE) unless your commander (Bn-level or higher) authorizes you to. You can be limited in your number of hours & where you perform your ODE. You are subject to recall and termination of your ODE, at any time, for any reason. You (or your employers) may not accept payment from TRICARE eligible patients you take care of. You may not use your military exempt DEA number to write for controlled substances at your ODE. You need to arrange for your own malpractice insurance to cover your ODE.
In my experience (Army only), and in talking with other colleagues (Army only), TO&E-unit commanders generally don't approve ODE. Some may be more amenable to their medical officer putting in a few hours at the local MTF, but many are downright resentful of their MO going out and getting supplemental income. The general opinion amongst line guys is that you should be taking care of soldiers/doing unit business with any extra time you have, like they do. And don't even think about requesting if your unit's medical readiness is not wired tight.
The strongest argument seems to be if you have perishable skills (like the interventional cards guy who is shoehorned into a Bde Surg position). As a GMO, I think you less likely to make that kind of argument than a BC/BE medical officer. One exception I've seen is if you're deployment-bound ED shifts may get approved more easily, but again, local MTF first. If you can get a more senior MO (such as the division or corps surgeon) to actively advocate for you, you may have more success.
TDA facilities seem much more liberal, almost capricious, in how much physician/surgeon ODE gets approved. Then again, they are a medical community, and better understand/accept the reasons for performing ODE.