Yes jimmarchi1 said it all, Soundsgarden Tone is based on the Amps, Kim Thayil uses only Pedals to boost the Amps and Live on stage he does switch Amps between a Mesa/Boogie Electra-Dyne Head and a Mesa/Boogie Dual Rectifier.
He does use Live MXR the Custom Audio Electronics MC-402 Boost/Overdrive and the MXR MC401 Boost/Line Driver just to push the Amps.
Live on Stage he replaced the Leslie with the Hughes & Kettner Tube Rotosphere MKII The Analog Leslie Simulator, of course not so good as a real Leslie.
For Experimental stuff he does use often the POG , a Dunlop MC404 CAE Wah Pedal and Technics playing behind the Guitar Bridge.
https://equipboard.com/pros/kim-thayil?gear=amplifiers
https://www.youtube.com/watch?v=uEl7u9Z3lK0
Jonny Greenwood is very different to Kim Thayil, Jonny uses a lot of Pedals doing very different Jobs and Amps more as a Pedal Plattform and not so much as a Gain Stage and I think you got already a good starting Point with the Fender also when I never played a Tone Master , but I think it can also do the Gritty Gain Stuff ?
The main Sound is by reverb and delay using the Roland Space Echo Tape Machine mostly over everything , Live sometimes the Boss Pedal Version of that and a lot of Electro-Harmonix Pedals like the Polychorus, Small Stone, Mistress , Freeze ... to build a very wide and big Sound.
For weird sounds Envelope Filter , Pitch Fork , Whammy , Tremulator, EarthQuaker Devices Pyramids, combined with Delay , Fuzz, Overdrive or even more Reverb.
And he does use Loopers , to layer more Guitars playing different Stuff.
https://equipboard.com/pros/jonny-greenwood?gear=effects-pedals
https://www.youtube.com/watch?v=PWKYP1oZY7w
https://www.youtube.com/watch?v=kTUgjC-1rfs
The Things both got in common is just some short weird Sounds created by Wah , Pitch .. combined with Gain , Fuzz, Delay and Reverb.
You Equipment is already a very good Starting Point ....
for Gain like already said its very different Kim Thayil uses extrem pushed Gain Amps,
Radiohead uses very different Gain Stages Fuzz , Overdrive , Distortion by Pedals like Boss Overdrive to push the clean Amp , Fuzz for heavy gain, Boss Distortion like Nirvana and so on ....
There is not really a so called 90ties Distortion Tone everyone did use, this is the Time where you still got very High Gain Amps used also in Rock, Grunge .... not only in Hair Metal and Trash Metal , but also Bands using a clean Amp more as a Pedal Plattform, but mostly high Quality Gain comes from Amps not so much from Pedals, when you play mostly Gain driven music , it is better to find Amp matching your Tone of Gain you like , when you play also a lot of clean with also effects are based on a clean sound , there is also a lot possible to consider as 90ties Distortion Tone it is not just the Boss Distortion , there are also Pushed clean Amps with a Tube Screamer or Overdrive, the Rat Pedal very extreme dirty aggressive Distortion, Fuzz Pedals on lower Gain settings , Kobain even used special Speakers in his Cabinet with a Power Amp on Stage, the Speakers were important to his Gain Sound , Smashing Pumpkins Gain was also on High Gain Amps and mostly on Big Muffs , it was not just the Boss Distortion Pedal there were a lot of different possible sounds in the 90ties with different ways to achieve a Gain Stage.
Also to say on a Record there is also Stuff used from the Studio , the Bands did not use on Stage , different Amps mostly to get more Quality to the Gain Stage and Live solved a different way , the Quality does not so much matter anymore and gets lost , to play Live is not a HiFi enviornment anymore.
You got already the MXR M69 Prime Distortion and the Plumes , giving you a good Gain Sound Pedalwise , on your whishing list is the Electro-Harmonix Op-Amp Big Muff Pi what makes sense, when you do not hit with that a Gain Stage you do like , it might be more to the Amp or the Tone you searching for is very Studio quality.
When you wanna switch to a High Gain Amp and do a clean and High Gain , keep in mind that this is your Gain Stage so you choose exactly the right Amp you want and mostly it will get very Loud , yes you can still put effects on it , but to switch Amps on the pedalboard or to get of the Amp Gain and using Pedals instead is always a painful job on the Pedalboard. The Pro is you got much better quality of the Gain, the negative is less variation of the Gain is possible or gets more complicated, more used to Music got Gain most of the Time.
The Clean Amp as Pedal Plattform gives you more different ways to use Gain by Pedals, but the Quality will mostly not be that great it will always sound a bit more LoFi-ish, you might also use more effects on the Gain like reverb and so on , but this is better when your music is not so much Gain Related all the time and more experimental and uses a lot more clean sections or Effects based on a Clean Amp.
Also to say when you listen to a Band you hear very loud Amps and the Whole Band, so you got the felling there is more Gain or Distortion to it, but you can not do this by just turning up the Gain Knob on a Distortion Pedal, the settings on the Pedal are mostly much lower then People think they should be, otherwise it gets lost in Distortion and muddy or too dark.