@egokun, while I haven't heard the HE4XX I own the HE400 (modded), HE400i, HE560, HD650, HD58X and a few other headphones, see my profile if you want to see what I have/had.
Since you have a decent amp (Magni3) you have a few more options open that I'd suggest:
HD700, yes many say its too bright but I think that those who say that have never listened to them since many that own them don't think they are overly bright and you Magni3 is a touch on the warmer side so it should be a decent pairing with the HD700, granted it might be more than a HD600 but used ones can be had pretty cheap these days.
HE560: this might have to be used but at $300 new I think its just a bit more than the HD600, it doesn't sound anything like the HE400 but for some the highs can get sibilant and don't get caught up with some who say they need a lot of power, its just a matter of finding an amp that pairs well and the Magni3 is s decent start but I like mine with the headphone output of my Teac UD301 (100mW) so power is not an issue for me.
If you can build (DIY) then I'd suggest a Nhoord Red V2 build, looks like a Grado RS2e but with drivers from Nhoord, very easy to drive but cost of parts put them in the $275-400 range depending on what you select (wood used for cups, headband assembly, cables, etc) but if you can't build then there are a few that will build them for you. If you want info on those that build let me know in a PM.
I'm not sure if you can find one but an Audeze EL-8 open might be something to consider, they were selling them off for $300 a few months back and to me its a pretty good deal, its like an improved HD650 to me, better sub-bass, no mid-bass hump, mids almost as good as the HD650 and highs that are cleaner and more extended than the HD650, but never sibilant.
FWIW I tried the M1060 and found them lacking/boring at times and also with a ringing issue at times, granted I only listened to two different stock units and it seems that many that like them mod them so its something to consider.
The last suggestion is a Grado SR225e with G-pads, while brighter than the HD58X, HE400i and HD650 they are much smoother so the treble is not harsh or sibilant to me. From there they are also easy to mod with dynamat, felt and sorbothane to tune the sound and these have a fairly noticeable impact on the sound from what I've found so far.
Good luck with your search and hope my suggestions help.
David A