The reason you won't find anything clean cut is because what Matt uses changes constantly and you can be very sure that he doesn't use the same stuff live as in the studio.
The Marshall is a good starting point. In generally, he doesn't actually use a lot of gain on his amp sound. This is so he can maintain clarity while playing chords. Keep your low end pretty tight (You can do this through a combination of reducing the Bass EQ on the amp, using an overdrive pedal (which I'll get to in a bit), and having a clean pick attack). He also has a very mid-range heavy sound. Again, dialling in some more mids alongside an overdrive will do this. In terms of Treble and Presence, try to add some nice high-end detail to the sound, but not too much as to make it sound harsh. I wish I could be more precise than that, but I don't know how your speakers, pickups, strings etc. are influencing the sound, so I can only be vague.
In terms of pedals, as mentioned, an overdrive will help add that bit more gain to the amp sound. Preferably running either your Blues Driver or the '59 Sound with minimal gain and maximum level (I've never ran either of those pedals into a Marshall DSL, so I don't know which one will work best. But have fun trying it!). You're basically looking to make the signal hitting the amp louder with a touch of distortion and with a more focused sound. Whichever one doesn't do this job best, experiment with using it as a distortion pedal for heavier moments.
Beyond that, it's honestly up to what sounds right to you. What I've laid out there will get you a solid rock guitar tone, which is really all Matt is using. There isn't really any particularly unique aspects about his base tone. It's designed to have the notes ring out clearly, get out of the way of the bass and have a touch of aggression.
I hope any of this is useful. If not, well ¯\_(ツ)_/¯