Beyerdynamic Amiron. They are very comfortable and sound great.
Going used will certainly get you more bang for your budget.
If you wanted to go with new you might try Hifiman Sundara or maybe entry level Grado.
I don’t recall what the headphone output of the SN3 is capable of, but something to be mindful of as I was pretty let-down by the headphone output oh my SN1.
Have a good read of the headphone pictures thread. Loads of good insight there.
The Meze 99 classic are easy to drive and very good value at half your budget
Great sound and looks, comfortable and closed back so you won’t disturb anyone - they meet my needs as an occasional headphone user.
They seem to be well reviewed across the web and at £269 new (prob half that if you get lucky on the auction site) a good quality bargain
Many will say do a demo as like speakers it’s a personal choice
I found the SIVGA Phoenix to sound good and are very comfortable.
Ideally you would want headphones with a low impedance for the SN3. I can vouch that a pair of Grado SR325 e work very well with the Naim integrated amps. As already suggested, cans from the likes of Focal and HiFiMan are also worthy contenders.
Comfort is just as if not more important than SQ.
Everyones head and ear positions are different.
Some really top SQ phones are really uncomfortable.
2nd shout here for Meze Classic 99.
Another vote for Meze 99 classics. Work well with everything Ive plugged them into
Comfort is important, I loved the sound of the Grado’s but they were too uncomfortable.
Couple of Meze headphones on eBay - buy at the right price and you can move on if not for you….
+1 for the Meze 99 Classics - I use a pair with my Nova and love them. They are also closed back so no sound leakage to wake your wife!
Meze 99 Classic certainly deserve an audition. Preferred or at least found them on par with competitors twice the price. This was a few years ago though.
A problem of advancing years
Grado RS series? You might get a deal on some ex dem or s/h in your price range. There are various models and some updates to the RS1 and 2 so old stock might be out there.
Grado are always engaging, fresh sounding and very enjoyable in my experience Although though I haven’t heard this range I have had three other pairs of different Grados over the years and there is a house sound that tends not to be a little bass light but with real pace and a very ‘open’ presentation. One to try maybe. Note they do leak sound badly if you are listening near your (sleeping) wife.
I don’t think you will be disappointed
PS - the drivers for Meze headphones are made in Ukraine so you are helping their economy
Are you looking for closed back or open headphones?
What music type of music do you listen to?