Heat is a big deal. If you do a closet you need to setup a fan or some other type of ventilation (vents at top and bottom to induce air flow, a duct off the flywheel fan etc).
I would certainly start with a muffler upgrade. Some of the 60 gallon, 3hp models have just a block of foam right in the head.
https://www.homedepot.com/p/Husky-60-Gal-Stationary-Electric-Air-Compressor-C602H/205389936
Units like that could really benefit from even a basic muffler setup. Something like this
https://www.amazon.com/dp/B00792G8PS/?tag=atomicindus08-20 (not endorsing this one, just that it has a snorkel on it) + about 2' of rubber hose off the snorkel can make a big difference. As I said, a similar setup on my 120V belt drive unit results in a 9db drop vs no filter at all.
Next you can look at things like putting vibration damping material around the tank. If you have a heavy mat this is something you could try for free. A harder job would be mounting the compressor and motor on a vibration isolated mount. If you look at the CAT compressors they have a unibody pump-motor but that lets them mount it with rubber grommets. Again, keeping vibration out of the tank.
Of course, moving the compressor out of the room is generally going to be the best.