Solve it.
Publish the code on github.
Do that in a few different languages.
Over 6 months figure you could cobble together 10 hacky minor (but working!) projects and 2 decent ones.
Do a decent job of a README.md which explains the problem the tool solves and put one of those nice animated GIFs showing the tool working or just screenshots so people can grok what the problem you're solving is and that you have a working solution (and nice screenshots will frankly discourage me from downloading your semi-hacky code).
Put link to github on your resume.
I would also say that if by "I could spend 6 months doing that" you mean that you want to quit your job and spend 6 months programming before you run out of money, I'd strongly suggest against that. Spend an hour or two a night and/or more on the weekends and stretch things out to a year or more. The stress of being unemployed and NEEDING to find a job is not going to be your friend here.