A few things that you might consider:
- Are you using the same module and/or audio output jacks when using either the monitors or headphones? If different, then perhaps it could be the module or specific jack being used for headphones. Some modules are just noisier than others, and unfortunately jacks can get worn out or damaged too.
- Is the same thing happening with different headphones? I recently had a noise issue with a pair of headphones, and then discovered that the 3.5mm to 1/4" adapter I was using on the headphone cable simply wasn't attached properly. My own fault, in other words.
- You could try using a linear power conditioner to reduce the noise.
- Do your bus boards have noise filtering circuits built into them?
Some of those may be less expensive to try out in comparison to replacing the power supply.
If you determine that it isn't the output module/jack or a specific pair of headphones, I would look into getting a power conditioner first, as that will probably be useful to have around at some point, even if it doesn't solve this particular noise problem. Intuitively, I also think that using a linear power conditioner would produce a similar result to switching to a linear PSU, but admittedly, I'm not an expert on these things by any means, and I very well could be wrong about that assumption.
Noise filtering bus boards would probably be less expensive than a new power supply, but also wouldn't necessarily fix the problem. I would recommend looking into getting some of those (if you aren't already using some with the power supply in question) if you think that a particular module is producing the noise or causing interference with other modules on the same power supply.
I don't have any experience with Meanwell power supplies, so I can't comment on any particularities of those.
I hope that some of that might help!