That dangling wire is a remote starter hook-up. Short length of loose wire with a female spade connector on the end that is covered by a plastic shroud, right? If you plug a length of jumper wire with a male spade terminal into that loose wire, then touch the other end to the positive battery terminal, it will crank the engine over. Very handy during valve adjustments and other work under the hood. Just make sure it's not in gear when you try it out...
I would avoid messing with the glow plug relay -- they are hard to find now and expensive if you break it by trying to alter it. If you are having hard warm starts then it means your compression is below spec, so eventually you will want to deal with that... but in the meantime, getting some glow when the engine is warm will work. A good way to do it is put an interrupt relay in the glow plug temp signal circuit, and then a switch on the dash to interrupt it.
So for instance, you would put a 5-terminal relay that is normally closed in line with the wire coming from the temperature sender at the rear of the cylinder head, right below the rear cam sprocket, and then a pushbutton on the dash to actuate the relay when necessary and temporarily open the circuit. Then, when the circumstances arise that you want the plugs to come on but they aren't, you can just hit that button and it will briefly fool the glow controller into thinking the engine is cold since it will see an infinite resistance signal from the temp sender. The controller will turn the plugs on for max glow, but after just a couple seconds of glow (should not need much when it is already partly warm), you can turn the key and the engine will start easily. Works like a charm.