The key is to not be so focused you are locked into a task and not so diverse you aren't an expert at anything.
Also knowing the market you are going into is a big plus.
I have about 15 years of mortgage experience. Did operations/secondary for a long time. Always doing computers as well. Banking was my part-time job so to speak while in college, became a career though and in the early 90's I was able to buy nice home in an great neighborhood pretty comfortably, but it wasn't my job of choice.
I went back to college, for my Comp Sci (4 year degree+ is important is looking at management) and started applying to large businesses.
I do support here on our Loan software, custom coding, reports, handle the hardware (desktop, server, tele/video conferencing) etc. I have proven I can do a lot of things extremely well and without much intervention since I know both sides of it.
Some support roles can never be outsourced. It depends how high up the chain you are supporting though. Desktop troubleshooting over the phone, sure. Custom code changes within an hour....not so possible. Being able to get a server up within 5 mins...nope.
Some companies can afford to wait though, we are not one of them...any down time for us brings down the whole Enterprise of 5000 associates, we have not been down ever...even during the 2 hurricanes that had about 75% of our staff without power for 1-2 weeks.