Dual Masters Necessary for Aspiring Chief Information Officer?

Hey!

I'm currently in my capstone course for a MS in Enterprise IT Management. I have an undergraduate degree in Intelligence Studies, and I'm currently Active Duty military. Both degrees are from American Public University System (a for-profit, regionally accredited school that has industry recognized programs, but sometimes receives criticism since they convene classes monthly to accommodate the demands of their students).

My intention is to work in that field, eventually obtaining a CIO role.

I have heard unsubstantiated rumors about those who aspire to be a C-Suite executive to earn a MBA plus a masters in their specialty.

Should I obtain a MBA, which would be a second masters for me? If so, would it matter where I earn that MBA? Should I only apply to those MBA programs accredited by AACSB? Or should I try my luck with a M7 knowing the expensive tuition costs, even after I use my military benefits?

I apologize in advance for the barrage of questions. I want to ensure I do enough research on this before I make any commitments.