What is the best way to request a new duty station?

Point to consider, the duty station you are requesting must have a need for your MOS. Not all locations are available to all MOS's. Locally, check with the Retention NCO. They have access to a program that lists vacancies by MOS (through Strength Management) throughout the army. They then use this information to "lock in" an assignment for a reenlistment contract. Branch Managers use the same system when considering reassignment requests or to fill TO&E manpower requests sent up by the unit.

Call them. I had never called my branch in 17 years of service when my father died, mother was alone, I wanted to be close to her. When I ask for an assignment in Florida he laughed at me. At that point I was straight up with him and little forceful in a nice way, I told him that I have never called and asked for anything of assignment branch before, always excepted branche assignments with no complaints, I needed this assignment. He then found me an assignment, curtailed my 3 year tour in Germany by 6 months with tour completed. Assigned me to MEPS as a Security Interviewer. Caveat being that after I reached my 20 year mark I put in for retirement, which was NO PROBLEM!

Made a deal with the devil!

Some branch managers would rather have your retention NCO/career counselor or first line give them a call or email. Some others don't really care. I will suggest you speak to your career counselor first. Better still, find out what your branch managers preference is before calling directly. You get a faster response when you call directly.