I am trying to find a 125V female plug receptacle with ground for a 12-gauge wire run in the 1960s, the ground wire of which is a much narrower gauge than the white and black conductors. The Levitons and the P&S's at the big box stores assume the three wires are going to be the same gauge, and so they don't get a tight grip the ground wire; it just slips right out. Is there a name for such a connector?
Don't add wire. Don't double the wire up, that won't hold, it will worm its way out especially if the strain relief isn't very good.
Dig into the gory details and you'll find that this connector is not listed for 18AWG wire. You can't use that connector for that wire.
What you're looking for is not big deal. Most corded sockets have screws that go all the way down.
One redeeming quality of big-box is they'll take anything back. Take it back. Grab the connector on the next shelf up that's 50 cents more. But first read the label and part if possible, and see if it's listed for what you need.
Better yet, go to a proper electrical supply house at a quiet time (after 9am). Tell the guy what you need and he will check the listings of each of his sockets til he finds one. Or, he can just sell you cordage and strain reliefs to replace your obsolete cordage.
Speaking of strain relief, that is important. That orange cheapie (a product of which I am familiar) has a strain relief fitting (and no doubt listed) to fit only one size of cable, i.e. One size of orange extension cord. Surely not yours.
If the grounding conductor is 18ga, that would create the risk of a ground fault which passes enough current to severely overheat the grounding conductor without passing enough to trip the breaker quickly. I would suggest that unless the breaker supplying the circuit is a GFCI (RCD) or AFCI, I would think it would be wise to use a GFCI outlet.
A GFCI or AFCI will kill the power if any significant current flows through the grounding conductor (at levels roughly two orders of magnitude below what would be required to cause overheating). Although codes generally require that all newly installed wiring have a grounding conductor the same gauge as would be required for hot and neutral, they generally allow the use of GFCI protection a substitute for a grounding conductor when wiring receptacles where a proper grounding wire is unavailable. Having an 18ga grounding wire in combination with a GFCI or AFCI should be almost as good as having a 12ga wire, and would almost certainly be safer than having an ungrounded GFCI. It may be good to talk to a local inspector for advice, since inspectors may have some discretion to allow deviations from normal code requirements when there is adequate justification.