11t-40t/42t cassette on 3x9 system

There are two things to be aware of here, three if you consider the hack.

First, it seems you have already considered max cog size. This will affect whether or not the derailleur can get the chain up onto the biggest cog. Technically, you need to be aware of minimum cog too, but you have an 11 which is arguably the most common for MTB. I assume this is for MTB based on the drivetrain you have described.

Second, you also need to determine max total capacity. This will determine whether or not the derailleur can take up all the chain slack when you are in your smallest gears. The post I have linked below does a pretty good job of explaining it. Once you know you proposed new maximum capacity, you can look it up on the shimano website for your (or any) derailleur. How to calculate the capacity of a rear derailleur

Third, let the hacking begin. If your numbers work, you are good to go. If not, whether or not you proceed really depends on you. Personally, I am the kind of guy that would go for it. I love tinkering with bikes, when things don't work out I get frustrated, but the process of what I learned, and the process I go through is worth it to me. It is my hobby, and I love it.

For others that just want something to work with minimal energy, it is a very different risk of your time and possibly money.

On two occasions I have made a RD work with a cassette that is technically to big for the derailleur. The first time was a Sram XD (I think XD anyway) rated for 42T and I got it working with a 46T just by cranking the B screw way out.

The second time, I was using a 10 speed system; a shimano derailleur designed for max 36T, and got it on a 46T cassette. This project took more tinkering. It did not work at first so I tried one of the derailleur hanger extenders, which made it work mostly. The shifting was not as good as I would have liked but it was for my son who was growing so fast at the time and I knew it was a temporary solution. It also may be worth noting that there was some previous damage in the derailleur from an attempted trailside re-straightening attempt.

Regarding hacking chain slack. My first hack I mentioned above was actually on what I call "Ben's personal 1.5X drivetrain". It was a 11-46 on the back combined with a 30T and a 26T up front but with no front derailleur. This bike was for my daughter who was very put off by the difficulty of climbing. The 30T was for riding around the neighborhood and just being a kid. Then, we would manually put the chain on the 26T cog if we went out to a trail where there would be some steep (from the perspective of a 10 year old) climbing. On the 26T chainring and the 11T cog, there was too much slack, but somehow it worked out, and it was pretty rare for her to actually be in that combination.

Again, if and how you hack is very much a personal thing. I love it and I have fun with the challenge of trying to get things to work when they don't. Others would consider it a loss if things don't quite work. Whether or not you embrace that risk is entirely up to you.

Good luck.

It's not clear why you would want to use an 11-42 cassette in a 3x drivetrain. 11-42 cassettes are typically used with 1x or 2x systems. With a 22 tooth chainring and 34 tooth sprocket you already have a pretty low gear ratio (0.65) - lower than a typical 1x drivetrain with a 30 tooth chainring and 11-42 cassette (0.71), and only a little higher than a 2x 36 tooth small chainring and 11-42 cassette (0.62).

Just to give you a definitive answer on compatibility: the two derailleur specs you need to know about are the max cassette sprocket size and the total capacity. Max sprocket sizes is obvious, total capacity is the ability of the derailleur to take up slack chain and needs to be greater than or equal to (difference in largest/smallest sprocket size) + (difference in largest/smallest chainring size). Your current setup requires capacity of (34-11)+(44-22) = 45.

You can find Shimano specs here: https://productinfo.shimano.com/#/, in the archive PDF spec sheets. The M592 SGS has max sprocket of 36, total capacity of 45 - which obviously precludes a larger range cassette.

If you used a Alivio 2x9 crank with 36-22 rings and a 11-42 cassette the capacity requirement would be (42-11)+(36-22) = 45, but your derailleur still can't handle a max cassette sprocket over 36 teeth. (Although, I've heard of people successfully using a derailleur hanger extension to accommodate a large range cassette.)