Using 30AWG wire-wrap wire as hookup

I use the 30 guage wire-wrap wire on pretty much everything and don't have such a problem. I even use it unshielded for interconnects. For unshielded wire to pick up hum from the transformer it has to be very close - almost right against it. An early headphone amplifier I built has the input wires going past the toroidal mains transformer within 1.5 cm and still no audible hum can be detected.

An enclosure will help, especially if it is thick aluminium, but you should be able to get it working OK out in the open first. There are lots of reasons for getting hum and I have found different causes at different times, but have never had to resort to shielding connecting wires.

Some possibilities are:
1. faulty wiring
2. ground loops
3. electromagnetic coupling to transformer at mains frequency
4. ripple on DC power supply
5. finger touching circuitry!

You could have incorrectly wired the signal return or power ground - re-check every connection point against your schematic.
Do you get the same hum in both channels?
Does it occur at all settings of the volume control, or only at max & min?
What are you using for a power supply?
Can you measure AC voltage on the DC power rails?
Does the hum disappear when you have a source connected?
Have you checked for sources of AC in the vicinity (e.g. soldering iron transformer, power supply, flourescent lights, etc)?
Your bodily presence will affect things too, so try raising the circuit up by putting it on a box or some books, so you don't have to touch it. Sometimes even waving your hand near a circuit can make a difference (the enclosure would help in this case).

OK, defining some terms might help here:
I usually take hum to be low frequency, at mains frequency or its first harmonic (50 Hz to 120 Hz). This will sound like a low-pitched HUMMMM or BZZZZZ, depending on the frequency and waveform.

Your reply mentioned hiss (HISSSSSSS), the sound of running water and white noise. This is a different issue to hum. It can be the result of the amplifier being unstable and oscillating at high frequency. This can certainly be affected by putting your hand near the circuitry (in my experience) and can be seen clearly on an oscilliscope. A symptom of oscillation can be the chip overheating because of all the energy being dissipated in the oscillation. Does the chip feel really hot to touch? It may pay to read Tangents notes on opamp stability. All his notes are very good in general and worth reading.

Some of the noise you hear is probably just what's present on the unterminated inputs, e.g. the pot movement. One thing I do is have a couple of RCA plugs with 1k resistors soldered from signal to ground. I put these on the input plugs of an amplifier if I am checking for such spurious noises with no source attached. This can help if the amplifier expects to have a low impedance source attached. At one stage I had various sorts of hum and hiss to track down. I also had to experiment with moving parts of the circuit in relation to other parts (e.g. large input capacitors). However, your amplifier is on a cicuitboard, so hopefully has been worked out by those who know better.
It may pay to search for threads specific to the Pimeta as each circuit has its own quirks and other builders may be familiar with this problem.