Why not just continue knitting in the round and increase for on either side of the "flap" portion, turn the heel, decrease back to the number of stitches you started with. It's often called a faux heel flap and is more commonly used on toe up socks but works for cuff down too. Usually the decreases are done at the ends of the needle holding the heel stitches.
I have done one pair with faux heel. Now I am thinking about ways to create a faux heel flap with most of the stitches rather than half of the stitches. I mean to create a heel similar to the one in this pattern:
Of course it is not too difficult to pick up the stitches in the traditional way... and it is safe and well tried. My first pair of socks were with a traditional heel flap. But it would be interesting to try something new and before spending hours experimenting I just want to get some input on the idea to see if someone already tried it.