Hey Otto,
There are those from the rated community that can answer your question with a bit more depth than I, but as a former 135 guy who was also involved in the management end of it (Safety) here's my .02 worth:
The primary Beechjet users out there are the fractionals. There are however, quite a few airframes in service both in air taxi and part 91.
Once you're qualified on your MWS, you're gonna have under 300 hours plus whatever you brought to the table as a civilian. Although on the low end of thing experience-wise, there's a good chance that with heavy networking and personal skills you can land a position in the right seat of someone's Beech. They're not going to save any money by hiring you, the insurance will require you head of to FSI or SF for training. (With the type rating and time it type, you may get by on a short course, which will save them a little bit of money.) The key thing will be you being a local guy who is eager and wants to spend some time with them. It would help greatly if the guy doing the hiring is prior military.
The biggest obstacle you'll have to overcome is convincing your employer that even though you'll soon be deployed to DirkaDirkastan, that it's still worth bringing you on board.
If your unit is in a large enough town, consider asking a Beechjet operator to pay for your short course in exchange for a discount on your services as a contract pilot. Then market yourself with other Beech operators in town as a contract pilot. Hit or miss, but it will put hours in you log book, and give you something to do other than bumming around your unit.
If you're successfully with the contract thing, I wouldn't go talking much about it to your bro's. You don't want any competition from the folks in your unit.
GOOD LUCK!