Whenever doing mechanical removal, especially with rhizome roots, you're not gonna get them all. Don't focus on that either.

They are like a battery, storing all their energy in the roots for winter and shooting back up during the growing season. If you remove the bulk, you're robbing the plant of a lot of battery, making it less able to pop up.

You'll have it come back. But in smaller amounts, which you can just hand pull as they do, eventually totally starving it.

There are certainly ways to find more roots, or times to pull (like rain or after it grows up) or other methods (like a foliar spray), but these all cost more time or include using poisons in your garden. Instead, just dig it, pull as it crops up, and see how it goes. It will likely be enough for less time and let you get into planting stuff.

Also bear in mind: your garden soil also has a seed bank ready to go, so once spring hits you're gonna get all kind of stuff growing there. If you're planning to grow your own stuff (food plants or native flowers), then you'll want to plant those partially grown after you dig, use a mulch for a year or so, and keep weeding. Eventually your planted plants will establish and inhibit other plant growth.