Hey stop it. You are giving away all the good secrets! LOL.
I use packing/shipping type foam, the kind like you would use in a Haliburton camera case and some of the old Shure mic cases.
Funny story only an old school bassist could understand. I had abdominal surgery about 18 months ago. When I woke up afterwards my feet felt funny like they were being squeezed and they were really warm. I slowly pulled one foot out from under the sheet and looked at it and I had these wacky looking blue foam moon boots on. They put these boots on real tight and they were fastened with velcro. Since I was in the hospital for abdominal surgery I was not expecting them to mess with my feet, but my very first thought was not about why they were there, but rather that I now had enough foam to last a very long time on my basses. This blue foam now resides on my 55 P-bass. Prior to this I was always snagging pieces of foam from shipping cartons and so forth for basses. This surgical boot foam is great.
I conceal the foam under the strings next to the bridge with a bridge cover. I usually like to try to match the foam to the bass color to make it even less obvious in case someone gets a glimpse...but heck I have a ton of blue foam from the surgical boots so I'm using it. It also reminds me how lucky we are to be alive. I've given some of this stuff to bassist friends too. This surgical boot foam lasts a VERY long time. It is high quality and quite dense.
MORE IS NOT BETTER. I have tried also loading foam both under and over the strings held down by the bridge cover and it just deadened the strings entirely too much. So after a five minute experiment I changed it back to under the strings only. Basically what I want to do is mute the harmonic tones, not mute the strings out much at all. It doesn't seem to bother the intonation very much at all this way.
With roundwounds the foam works well, but with flatwounds it tends to be overkill. So on flatwounds I have found that about a 3/4 to 1 inch strip of FELT from a craft store overlapped twice around all the strings adjacent to the bridge. The craft store here has about 100 colors of felt in 8x12 sheets for 35 cents each so you can get close to your bass's color to make it less obvious. The felt only touches the strings. I Scotch tape the end of the felt down to the underlying felt layer and then slip the entire felt wrap around the strings until the tape is on the bottom. Works well.
Dang another secret out of the bag.
It is amazing how much difference the right material makes. I may try to steal a diaper somewhere and try that. You never know.
Think of all the money people spend of pedals, rack gear, strings, pickups and other gimmicks when sometimes the sound they are looking for comes from the judicious use of free foam or 50 cent felt.
Most old Fender basses used a narrow strip of real rubber weatherstripping type material under the bridge cover. The weatherstripping rubber eventually deteriorated, became ineffective and got hard and was removed. You'll see the discoloration on the underside of bridge covers sometimes where the rubber once lived.
Also Fender briefly used a more sophisticated felt pad mute system on the Jazz basses. This had individual arms with felt pads that pushed down on the strings to mute them. These were somewhat adjustable as far as the pressure was concerned, but they were quickly abandoned and Fender back to the weatherstripping.
My hat is off to anyone who can consistently mute using their hand. I get either too much or not enough mute that way.