I have double stitched a few cases that the old cotton stiching fell apart around good shape leather.

I first used waxed white cord from walmart. I wandered into Film Plus in west end Toronto one day with one case done this way.

Kat was pleased to see the old folder back in service, but was a bit horified with the ugly cord.
She pulled out a big spool of the right heavy black cord and reeld off a few dozen metres and gave it to me, and admonished me to not restore with white cord again.

By the way, if the leather is stiff, while apart, saturate the tanned side with lanolin paste and then wipe off the excess after it has sat for a few weeks. That makes the old leahter supple again so it will not crack.