01 license allows you to do any electrical work in Washington, including 06 work. I don't recommend doing the 06 apprenticeship if you want an 01 license. You'll spend a lot of time at a lower wage, then when you decide to jump into the 01 apprenticeship you can apply a lot of those hours towards getting your 01 license but you lose a couple years of experience as an apprentice doing 01 work (which generally has almost no overlap with low voltage work in commercial).

Plenty of guys do go that route but I think it tends to stunt their growth into a journeyman wireman. The only reason I could see going that path is that the competition to get into the 01 apprenticeship is much higher and could take you much longer to get in. Experience as an 06 will help push you through the line.