Hi
The fancy name for what you are playing with is an FPC cable / connector. There is a multi layer problem here:
1) The interconnect cable is a bit tricky to do.
2) The connectors are a bit weird
3) If you simply buy the cable, it tends to be connector speciific
Simple answer is to pick both the cable *and* the connector at the same time. If you already *have* a connector on one end, you need to get some pretty good data on it. Might you get lucky? Sure, it's quite possible you will.
Once you find a connector, you will likely have to do up a custom part for it in your layout program.
Bob