I agree with the guys who say single/0, that's what I do on part-time jobs as well (except the years I knew I was getting huge refunds). Your full time job takes up all of your non-taxable income, so you're really taxed on every dollar of your part-time job. I'd revisit the full-time job's W4 if anything. Especially if your PT job is for "spending money", you really don't want to spend money you might need for taxes later on.
probability yes change it to zero or single
I would file o on that as well. The extra 6K could push you in a whole new bracket.
You didn't follow all of the instructions. When you have more than one job, you must use the worksheets that accompany Form W-4 to ensure that the proper amount of tax is withheld. The calculations can be somewhat messy, so the IRS has set up a Withholding Calculator that's linked on their home page. Use it.
If you are claiming 0 on the first job and only earn $6k on the second one, you *might* be close at filing time. Either use the worksheets or the Withholding Calculator to be sure.