Re: Pedal for a dirtier and thicker lead tone?
I have a Mustang III and my favorite overdrive pedals to use with the Mustang are included in the amp.
The Overdrive (Fender Overdrive) and the Green Box (Tubescreamer) are my two favorite OD's included with the Mustang. Both sound terrific.
I own lots of pedals, including a Klon KTR, but I don't use them with the Mustang III.
The "pedals" included in the Mustang are all I need.
I use a PRS with Seth Lovers, a PRS with Pearly Gates and a PRS with Duncan 59's and the Greenbox or OD to get great Santana tones: fat and thick with terrific sustain.
I think the OP needs to get to know the amp better. Because it's all right there in the amp.
The 57 Deluxe amp with either the OD or Greenbox (Tubescreamer) gets a beautiful soloing tone.
As does the Deluxe Reverb model when it's goosed by the OD or Greenbox.
I don't use the Marshall, Vox or Mesa models much.
I prefer the smaller 20 watt Fender amp models, but when I'm going for a Clapton/Cream or Santana or Peter Green or ??? overdriven soloing tone, I always use the Fender OD or Greenbox to overdrive that amp model.
When I'm playing clean I like the Twin Reverb model.