Hum decreases with tonearm position. Huh?

To all of you experts:
Help, please. This is making me crazy.
I've got an annoying hum coming when I use my turntable. Here's the setup:
Rega P3 2000 with Elys cartridge;
Ray Samuels F-117 Nighthawk phono preamp. (This is a battery-operated unit.);
Wyred 4 Sound STP-SE preamp;
W4S mono amps (I forget which models. Not the lowest, but not the highest.);
Celestion SL-700s.
Everything is plugged into a PS Audio Power Plant Premier.
The preamp is connected to the amps with W4S XLR interconnects. The TT to the Ray Samuels is via the dedicated cable on the P3. I've tried various interconnects from the phono pre to the preamp with no difference.
Here's the story: There's a hum, which can be heard with the volume set even fairly low, at even low listening levels. But it is loudest when the tonearm is in its resting position, lessening as I move the arm toward the spindle.
I've tried moving the preamp away from the TT, but that doesn't change anything.
The hum isn't so loud as to interfere with listening to records, but it is loud enough to be very, very annoying when I'm, say, getting ready to spin a favorite platter. Or when I'vm flipping one over.
Any thoughts about what might be causing this? My cartridge is getting a bit long in the tooth -- I've been thinking about an upgrade -- but I don't see how that could be causing this. And I'm not really excited about dropping some real green on a new cart until I've got this hum thing resolved.
Is it odd -- it seems so to me -- that I've got hum with a battery-operated phono preamp?
Thanks so much for any help you might be able to give me with this.
-- Howard
That's why Rega P3 lasted in my system only one month. The hum is originated from motor under the glass platter. There upgrade options to replace a noisy motor plenty on the web or just think of replacing it with better one for money.
Thanks, Marakanetz, but if it was the motor causing the problem, wouldn't the hum get louder, not softer, as the tonearm moved toward the center? In my case, it is loudest when the arm is in its resting position, decreasing as I move it toward the spindle.
Marakanetz is entirely correct. I made a plate for the top of my Rega P3 from Mu-metal, which stopped the hum, but that was a lot of effort. getting a different table is the better solution.
Does this only happen with a P3? I have an RP-3 that I bought in May and there is no hum at all.
Agreed, The Rega is lower quality than the rest of your system. It's a budget entry level piece made for starving Brits who want to play records. The list of upgrades is endless, and by the time you pay for it all, you could have had something good.
It should be the other way around. The source is the most important piece. If you don't have a quality signal, it doesn't matter how good the rest of your system is.
Hodu, the spherical distribution of EMI is somewhat similar to fire where the focus(the hottest spot) isn't at the 'tip of the match' but a little further.

Polk432, don't know about RP3 which probably has all of the possible upgrades since P3 and priced accordingly. P3 was budget oriented table that was not performing great for the budget for various reasons listed here plus reasons that are not listed here.

Back than, there were quite large list of vintage tables that could easily outperform P3 in terms of performance for the money paid.
The motor on the Rega P3 may or may not be of low quality, but quality has nothing to do with the hum. The problem is that the glass platter affords no shielding against EMI emitted by the motor that may or may not interact with the cartridge. As someone else wrote, installing a shield will and did cure the problem. Another alternative is to ditch the glass platter in favor of a metal one that in itself will serve as an EMI shield.