Checking a puddle of solder itself won't really help much.
If you want to see that one side of a wire is connected right you can put on probe on the end of the wire you aren't checking and the other on whatever it is the end you are checking is supposed to be soldered to. If the end you aren't checking isn't accessible, you might have to go another level back and check through a component. For example, if you want to see if the ground wire of a pickup is soldered to a pot well, you can put one probe on whatever it is you're soldering the wire to (I wouldn't touch the solder joint itself) and put the other probe on the end of another wire of the pickup (the hot wire, for exmample). If it's soldered well, you should get some sort of resistance measurement that isn't super high (you can touch the end of the wire you are checking to see what the value should be). If it isn't soldered well you'll get a value much higher than that or it will just show whatever it shows when the probes aren't touching anything.