You can try using the IRS W-4 calculator
https://www.irs.gov/individuals/tax-withholding-estimator
Remember the result will likely be your total recommended withholding. You could subtract your 2021 withholding from your full time job from this total to get the additional withholding recommendation.
I know you said you don't want to mess with your full-time job's withholding, but that may be the simplest way to deal with this.
If you want to, you can also choose to file quarterly estimated tax payments. But that may be a bit more cumbersome than changing your withholding.
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