When I turned both gain and trim knobs fully clockwise with nothing connected to the input,
I listened to the clip. I can hear some hum and RF noise. At extremely high gain, there will be white noise. At max gain, we're talking about 72dB, which is 4000x amplification. Any little noise, will be amplified 4000 times! So you can see how it is very important that construction and wiring be as clean as possible.
Your hum seems loud... but I also see that your power transformer is near the preamp. See some tips below:
Also, how did you record these? From the preamp out, did it go to a Mic input of your audio interface or the line input? Is it a balanced or unbalanced connection? Is there any "gain" being applied by the DAW software or audio interface?
Here's a sample of my 72dB, 150ohm terminated input, well grounded case, 1u rack, PSU and transformer far from the preamps... sound clip.
Some suggestions based on the photo you sent.
1. Ideally, you want the power transformer and AC lines as far away as possible from the preamp. Otherwise, any slight hum will be amplified... especially at 4000x.
But since your case is small, I suggest tighter and cleaner wiring. For example, I see the DC power supply wires (CABLE4) going to the pre almost touching the power transformer. There's plenty of room to reroute that... move those wires away.
2. Did you ground your case? I mean truly grounded? It looks like a painted case and I wonder if you removed any paint to ground it. Otherwise, the paint will act like an insulator. Did you ground the XLR tab and pin1 on the XLR jack too?
My hunch is your case isn't well grounded...
3. In real life, you won't be running the pre with the inputs hanging. There will be a mic connected to it. Have you listened for noise with the inputs terminated with a 150R resistor? to simulate a mic connected to it? This should result in a lower noise floor compared to the above recorded sample. (Also, take note... your above recorded sample is 72dB gain!)
4. The trim knob should be set at max clockwise... that's the 0db mark, i.e. no trimming done. Only at high gains where you want an "in-between" setting should you turn that trim down a bit.
If you really want the best performance out of the preamp, try a bigger case with the PSU section, as far away as possible from the preamp. Otherwise, hum and noise brought by the AC lines will be to near proximity to your preamp.
5. Have you tried the rotating toroid trick? I.e. loosen the bolt of the transformer, and turn it around slowly and listen for the sweet spot where noise dramatically goes down. Sometimes a few degrees of turning have a big effect on lower noise floor. WARNING: This needs to be done with power ON so you can listen to it... otherwise, to be safe... turn power on/off, adjust toroid, power on/off listen... repeat.
6. I personally don't use shielded wire for the XLR in/outs. Instead I use stranded wire and twist it tight. In my experience, it performs better than using shielded wires for this short run. Experiment if the noise goes down if the wires are right against the case floor, or elevated.
Hope this helps... there is no ONE magic thing to do to ensure a low noise build. You need to be meticulous on every aspect of the build as every little thing can add up to a big noise. Hope the above pointers help you.