Sometimes cables fresh from the factory are the worst for schmutz. There can be like this oil stuff on the connector, that is a residue from the manufacturing process.
And what is worst is when you have the oil get from the jack into the socket, because it is harder to clean then. The way to clean that out is to first clean the jack with alcohol, then plug the jack in, jack it in and out, and then clean the jack again. Repeat until you get a solid connection every time you plug in. Your using the clean jack to wipe out the socket.
A totally different topic... if you crank an amp up and the hum gets relatively louder than the music, it can be a sign of having an electrolytic power cap in the power supply going bad. What happens is that the cap is weak, and at high current draw of the amp, the weakness shows up as being significant, but at low volume the cap is able to keep up with the current demand. What was a hefty 4700 uF cap has degraded to a 500 uF cap, and it is not able to smooth out the 60 Hz ripple of the AC power supply when load current demand gets high. There isn't much you can do about a problem like that, the cap will progressively degrade over months or years, and eventually fail altogether. The cap has to be replaced.
If the hum is coming in through the signal path, the relative volume of hum to music volume will remain the same as you crank up the amp.