Tax Year Prior to 2020: I started a part-time job that will supply 1099 income in addition to my W-2 income from my full time job. What is best way to structure tax payments on the new income

For the 1099-MISC income, you are considered to be a self-employed contractor and will file a Schedule C as a sole proprietor.   For federal taxes, there are two amounts that need to be added together for your estimated tax payments.  The self-employment tax rate of 15.3% plus your ordinary income tax rate (which depends on your taxable income and filing status).  The following article has easy to read tables for this amount http://taxfoundation.org/article/2017-tax-brackets

How you figure out your estimated quarterly tax payments amounts is to take your gross self-employment income minus expenses to get your "net" taxable self-employment income.  Then, multiply that by the 15.3% self-employment tax rate.  

Then, look up the rate on the charts for your annual total taxable income (in the link above)  and multiple that by the percentage for your total income you expect for the entire year. Add those to tax amounts together.