Ask HN: What can I do after graduation, but before I'm qualified for positions?

After reading this again, I can see how it might look like a brag post. I swear that it's not, I'm legitimately seeking advice.

I'm coming up on my senior year of college, but only discovered my love for computer science late in my academic career (I'm a math and biophysics double major). Because of this, I've only had about year of true programming experience.

And although I'm proficient in C/C++, Java, and Python, and have had two really interesting research positions (writing applications for HPC/supercomputers and hypervisor security research), I still don't feel like I have a whole lot of the experience people look for in programmers.

So as graduation comes up in the Spring, I'm at a dilemma. I need to either get a job, or go to grad school. But can anyone offer some advice on jobs I could look at that would help me continue to learn programming? Or, alternately, has anyone went on to get say, a Master's in Software Engineering, and could share their experiences about it?