I've just been an audiophile for a long enough time I guess. The HD600 is great, it is endgame for many people and routinely beats headphones costing several times as much. Grados I quite dislike, they have really high distortion in the bass and have audible ringing at around 2-4khz. But if you like them, you do you I guess. The HD600 is in a unique spot where there are really no headphones out there that are a direct upgrade across all areas. Its main weaknesses are bass extention and dynamics, something like an hd800 or focal utopia (both of which are pretty different from one another) will beat it squarely in both of those areas, but those two headphones don't have the highly flat diffuse-field tuning of the hd600. If you like the hd600 and grados, you probably prefer a drier tuning, so a non-exhaustive list of some headphones you should try might be:

Focal utopia - These are somewhat v-shaped, extremely resolving, possibly the most "dynamic" sounding headphone, very narrow like the hd600, makes hd600 sound veiled and compressed in comparison.

Sennheiser HD800 - Definitely a polarizing headphone, bright in the lower treble but lacks some presence due to a scooped upper midrange @ around 2khz, so the opposite of grado in that regard, super enormous stage (second only to K1000) but imaging is not quite as precise as utopia or K1000. Great bass with minor rolloff, makes any hd6x0 sound wooly and imprecise. Extremely resolving.

AKG K1000 - Their presentation is a lot more like speakers than headphones. Very unique and incredibly "live" sounding. They are forward at 2khz but not as much as grados. They can sound alive and dynamic, or washed out, dead and veiled, all depending on the angle of the drivers, so you really have to adjust them perfectly to see what these can do. More resolving than hd600 but not as much as modern flagships like the two above. They have the widest stage of any headphone, it really has to be experienced. It's as if there are actually palpable instruments floating in the space around your head. Amazing realism, but highly uncomfortable and not ergonomic in the slightest. Ones with serial #s below 5000 are the most desirable as these are the "bass heavy" versions which were made before a factory revision that increased the primary resonance of the driver from around 40hz to around 60hz if i recall correctly)

Mysphere 3 - Modern K1000 successor, designed by the same guy who designed the K1000 in the late 80's. I have not heard them but they are said to improve on comfort, resolution, bass extention, and driver sensitivity.

I would be cautious with planars. When they aren't tuned to sound like listening to music through a sock (audeze) they often have a weird habit of sounding simultaneously bright and veiled, which isn't exactly pleasant. Not sure why that is from a physics standpoint. the diaphragms are usually held to high tension, so it might have to do with standing waves forming on their surface at high frequencies? Idk. What planars excel at is mostly their unmatched low frequency capability, with bass extention flat to well below 10hz with a perfect seal. Good planars tend to have a sense of huge weight and solidity but without the bloatedness and blurriness of bassy dynamic headphones. My favorite planar is the hifiman HE-6, but the driver has a lot of ringing in its stock form and needs to be modded to reach its full potential. Also the drivers have a nasty tendency of dying, and I don't think the manufacturer can replace them anymore. Like the K1000, they are also extremely insensitive. I unironically use a speaker amp to drive these. If what you want is an upgrade in terms of resolution, planars probably aren't where you want to look, as they usually have some degree of haziness to them that you don't find with dynamic drivers.

Any stax - These are electrostats and require their own high-voltage amps. There are warm stax (sr-007, sr-L700) and bright stax (sr-009) but generally speaking they tend to be highly resolving but dead sounding and lacking dynamics. This might be due to the limited excursion and low driver mass, again not 100% sure why. Also they tend to have an upper mid shelf (wider than the narrow 2khz-ish boost present in AKGs and grados) from around 1000hz onward that lends everything a kind of plasticy timbre. They are really cool from an engineering standpoint but in my opinion overpriced and not that musically engaging.

For your preferences I would stay away from: Audeze (veiled compressed dead sock), non-flagship hifimans such as he-500 and he-560 (veiled and hazy to the max), anything mrspeakers (grainy compressed dead sock, also veiled), ZMF (legitimately nice sounding boutique dynamic driver headphones, but definitely colored and significantly warm of neutral), the senn hd700 (truly an abortion of a headphone), the hifiman HE1000 (veiled, warm, wide, sounds like listening through two cellphone speakers placed 6ft away from your head), and any fostex dynamic drivers (highly dynamic and v-shaped, opposite of hd600)

Most of these are really expensive and/or unobtanium, but as I said you have to do quite a bit to beat the venerable HD600. It would be crazy to buy any of this stuff blind, I would recommend finding a local high-end audio shop where they will let you sit down for a few hours and listen without interruption. Head-fi meets are also a great opportunity to listen to a bunch of gear and meet new people, I'm not sure how often those occur anymore though. Cheers!

I appreciate the detailed response!

What kind of music would you recommend listening to to evaluate headphones in a shop? I plan to grab my HD600 and Grado and compare them to HD800. Aside from Sennheiser and Grado, my local shop lists the following: https://www.missionaudiovideo.com/catalog/headphones-and-wir...

I was told the high impedance can be an issue when listening from a desktop/laptop/phone (as I normally do), so I was actually considering HD660S or even HD598/599. What do you think?

What about in-ear models? I hate the fit of the stock iPhone earbuds (they just don't fit me at all), but provided I find those that fit well I would like to own high quality earbuds. What's the difference in terms of sound between in-ear and around the ear designs?

Listen to anything you like that's well mastered and in a lossless format. Acoustic music recorded in natural spaces tends to reveal the capabilities of a transducer better than synthetic or overly close-miced recordings do, in my opinion. Classical and jazz tend to be most useful here because they tend to escape the dynamic range compression that pervades all of pop and electronic music these days, but there is well-mastered music in any genre. Any music you are super familliar with will basically tell you all you need to know really. A few albums I like to listen to when evaluating gear are Pat Metheny Group - The Way Up, Aphex Twin - Selected Ambient Works 85-92, and Miles Davis - Bitches Brew (not the remaster).

I don't think any of those other cans listed online will challenge the HD600 at all, I know the nightowl and se-master1 are both notoriously bad. But if they have an HD800, definitely go have a listen. It is a polarizing headphone and whether you like it or not will tell you a lot about your own preferences.

High headphone impedance is only an issue if the source you're listening out of has a high output impedance, if the output impedance of the source is high enough, the frequency response will be altered and there will be a FR bump around the primary resonance of the driver (usually around 100hz in most dynamic drivers) due to a lack of electrical damping. A dedicated headphone amp is always best, there are a lot of good ones out there these days that don't break the bank.

I'm not an IEM expert, I've never really felt the need for them and have never found a comfortable pair either. Over-ear designs should have more spatial realism as the sound field interacts with your whole ear (research HRTF and pinna transforms if you want to learn more) compared to IEMs which fire directly into the ear canal and so skip the HRTF entirely.

What do you think about Apple Homepods?

Haven't heard it. I know a lot of people were hyped up when it first came out but I don't really know if it delivered or not. It's supposed to be well DSP'd so I wouldnt be surprised if it beat most of the speakers in its price range though.

I'm thinking about getting them as desktop speakers, as an alternative to using headphones at work, because my ears get hot after a couple of hours, even in HD600.