Help diagnosing a buzz (Fender Telecaster)

If the problem was a noise increase when touching the strings, that would signal a problem with the circuit.

Shielding *may* help reduce the noise. If you shield the pickup routs and control cavity, be sure to add shielding inside the drilled bore to the output jack too.

Some environments are just simply noisy.

Fender single coil pickups, even so-called noiseless versions, sometimes do make unwanted noise. The fact that it goes away when playing (fingers on the strings) is good! One thing you can do is keep edge of the palm of your picking hand against the bridge plate. If you are a right-hand muter, this is a simple fix because it's already habit for you. When not playing, get into the habit of sliding the palm on the volume knob to kill all signal to the amp. Playing single-coil pickups sometimes requires these sorts of techniques.

You might want to pop out the amp chassis and look in the cabinet to ensure there are no "open" (unshielded) areas around the chassis, e.g., inside the top panel of the amp cabinet. If there is, use aluminum sheet or copper tape (staple it down; don't rely on just the adhesive) and make sure it is all in continuity with ground when the chassis is reinstalled.

IMPORTANT: If you are not familiar with amp guts and discharging filter caps, do not open it up because there can be residual voltage even in an unplugged amp that can kill you to death.

Same goes for the shielding tape in your guitar. When done installing the copper tape, test across all the shielding in multiple locations with your meter to ensure you have continuity throughout. There are copper tapes on the market that claim to have conductive adhesive, but sometimes that is not the case. I always add a dot of solder between any two pieces of copper tape when shielding a guitar. And do not be concerned with ground loops or 'star grounding' in your guitar. The circuit is passive and the voltages and currents are too low for differential ground potential to occur.

Amps, however, are different due to higher voltages/differing voltages/variable ground potential and must be star grounded to keep noise at a minimum.

Have you tried that guitar through different amps to see if it's quieter? Keep in mind that different amps have different gain characteristics, and gainy amps will usually be noisier strictly due to how they operate.

How do you like that Orange rocker?

I so wish it were made in England instead of China, but it has a very warm clean sound that breaks almost immediately on 0.5 and 1 watts, but has some nice headroom at 7 and 15 watts. The "Dirty" channel spans everything from mild overdrive or classic blues/rock tone to full-on heavy metal. Very versatile if some flavor of "rock" is your thing. It also has a tube buffered FX Loop which seems a premium at this price point.

As a bit of an update, I took the guitar apart and checked all of the grounds and wires and everything seemed kosher. I also took the time to fully shield the pickup cavities, around the tone controls, and extra shielding on the pick guard. It sounds a bit quieter, but the hum is still there when I'm not touching metal on the guitar or amp.

I also checked the ground of the outlets I'm using and they're perfectly fine.

I think it may be a combo of lots of EFI/interference in our home plus some wiring that Fender could've done a better job on (more shielded wire? more grounds?), but it's acceptable at this point. The guitar plays and sounds so good otherwise. I also have an E-H Hum Debugger I'll run first in my chain of effects off my guitar to see if that reduces it a bit, but I'm good with it now.

I just wish Fender put the same effort into this that Gibson does (or at least did for 2011-era models). Especially since the price point of the American Elite/Ultra is much higher than an SG Standard...

I had an unexplained noise problem at a gig and it turned out to be caused by my cellphone in my pocket.