Love / Hate relationship with vinyl?

My suggestion (no big revelation here, I know) is to look at your phono stage (which is) and see if the noise is coming from there. Also check your total system gain to see if perhaps the head-amp plus phono plus pre plus amp has too much overall system gain. If you can't eliminate it, you have four options:
1. Try an SUT. These are typically very cart specific as to the total loading you end up with, so do your research before you buy/audition. However you may find that you are also sensitive to ringing from an SUT.
2. Try a phono stage o /full function pre that has sufficient gain for the Kleos. There are several out there that will do the trick. For example, the ZYX Artisan was dead quite with a LOMC in my system.
3. As you said, try a higher output cart. If that was the route I was going, I would probably avoid an HOMC as the added mass from the additional coil windings would tend to counteract the low moving mass of an LOMC, which is one of their advantages. And don't discount an MI cart like those from Soundsmith.

First thing I would try is the MC input w the lower gain option. It may be sufficient, depending on the line stage and amp gain you have and the efficiency of your speakers. KAB has a gain calculator app that lets you input your cart voltage output so you can figure out how much OVERALL system gain you have, which is really the relevant variable.

When I move (or even touch) the cables going to my headamp, the level of hum changes.

Bingo! or maybe bingo ;-) I'd try different ic's to/from the headamp and also check to see if the RCAs on the headamp are secure and properly grounded.

That's good news. I have been told that cable effects can also be stepped up w an SUT so it's not clear exactly how it contributed to the hum, but who cares! Enjoy.