Re-reading the original post, it appears you already purchased a replacement muffler correct? Were you asking what additional can be done to help with sound attenuation? I originally thought you were asking how to repair the original muffler that had the foam material that had deteriorated. It doesn't look special - like the same type of foam material used in window AC filters and the like that you can usually find at big box stores like Home Depot. It can't be too restrictive because you still need significant airflow through the material - think of it more like a diffuser to redirect the airstream on exit out the exhaust. Not sure if that muffler opens up for doing this or not though. For those that haven't seen these vacuum mufflers and wonder if they work, the anecdotal video below may help. This was taken on a Ridgid shop vacuum using a handheld sound meter with the muffler on and off. He got about a 5dB drop in noise level using the muffler. Based on my own past experiments I would totally believe this. A ShopVac is NEVER going to be quiet unless you put it in a soundproof cabinet, which some have done and there are videos of them. However a 5dB drop is significant and noticeable and worth doing. A good portion of the noise of a ShopVac is the high speed, turbulent air exiting the exhaust port. I did some experiments with mine where I attached an 8' vacuum hose at the outlet (exhaust) and placed the end outside of my garage. I lowered the garage door down on the hose. Even with the garage door cracked open a few inches, simply directing the exhaust hose away from the shop outside resulted in a significant reduction in noise inside my shop. I don't remember the sound meter measurement, but I remember I could tell it was noticeably quieter and the resulting noise level was less harsh inside my shop so the quality of the noise changed as well. The hose had sufficient dampening to minimize noise transmission back into the shop. Hope this helps...
https://www.youtube.com/watch?v=KwBOtHepqfY&t=1s