Okay So I've done a bit of looking around in the headset. It seems that there aren't any actual screws that tighten the swivel but only ones that hold the headset in shape.
These are a few images I took of the area but it doesn't look like I can do much to tighten it (Also apologies for poor photo quality. It was from a gen 6 ipod lol) https://imgur.com/a/YJygRNQ