The knitting that you want to change. In this case, I wanted to change edge of this Gatineau Hat from 2/2 ribbing to garter stitch.
Picking up a round (or row). Using a spare knitting needle, pick up a row of stitches. You don't have to use your gauge needle; a smaller needle will make it easier. You can also use a blunt yarn needle threaded with some scrap yarn. When you pick up, try to pick up the right (as opposed to the left) leg of the stitch. This will mean your stitches are positioned correctly on the spare needle.
The stitches, safely picked up. After you've picked up all the stitches, do a quick spot check to make sure you haven't missed any.
This will feel wrong, but if you need to undo knitting below the needle, you will need to cut away the unwanted knitting. If you try to unravel starting at the cast on edge, you will soon find that the yarn is locked at each stitch. (Unlike unravelling from above the needle, where the stitches undo very easily.) Very carefully, cut the row or round above the picked up stitches. (I'm "reading" my knitting in the direction I'm going to knit, so when I say "above" I mean 1 row towards the unwanted knitting. Don't cut the knitting you want to keep!)
Freshly cut. Here's my hat after cutting. You can see that the cutting leaves messy little yarn bits.
Cleaning up. Clean up the bits of yarn around your stitches. Then you will have a fresh round or row, waiting to be knit. Simply join your yarn and start knitting! You can work directly from the spare needle even if it is smaller than your gauge needle.
Reclaiming. After removing the unwanted knitting, you can unravel it and wind it into a loop. It won't be usable in this state (it's too kinky). Tie the loop with some scrap yarn in a few places to hold it together.
And then soak it. Allow the loop to soak in tepid water for at least 20 minutes. Then remove it carefully and blot it on dry towel. When the loop is dry, wind it into a ball.