When you convert a portion of your IRA to a Roth, a proportional share of your basis is allocated to that distribution.
For example, if you open a traditional IRA, make a $6,000 contribution and then convert it to a Roth, the entire distribution will be tax-free because your basis is $6,000 out of a total $6,000 in your IRA.
If you open a traditional IRA, rollover a 401(k) worth $66,000, and contribute $6,000 in after-tax dollars, your total value will be $72,000 with a $6,000 basis. If you then convert $6,000 to a Roth, 1/12 or $500 will be tax-free and the remaining $5,500 will be taxable. The remaining $66,000 in your traditional IRA still has a $5,500 basis.
For more information, the IRS provides worksheets on calculating the taxable portion of your distribution. It does not matter if you conducted the conversion prior to rolling over your 401(k), as long as they were done in the same year. It also does not matter if you open one IRA for back-door Roths and another for rolling over your 401(k). The basis is calculated on the value of all your IRAs.
