One of the best programmers I know (became VP of Engineering at a major company at one point) was an English Lit major with no formal CS training.

He started off consulting and worked his way up through a lot of hard work. Point is, I actually think it would be faster to do this (assuming you're willing put in the effort) than to aim for a master's program with the hope the end result is a better opportunity. After your first programming job, experience and recommendations matter much more than a degree.