Well plenty of us use dirt pedals for varying amounts of... dirt. If you're using a tube amp and want more breakup than normal volumes will give you, you might be able to use a simple boost pedal so slam the amp harder. I don't have any personal experience with that, but I've read plenty about it. I use a Damnation Audio MBD2 for my dirt needs and it's wonderful. Very flexible. I've got it down to just adjusting one knob for going back and forth between a GK and the B-15. Sounds great on all. Really tightens up the signal as well.Not bad if fixed right. Slimy doesn't work.
I wouldn't mind matching the power tube breakup and compression found in one of my Ampeg heads, but it's way too loud to be usable. My Sanamp VT will allow blending in some distortion, but it just doesn't seem to fit well. But it's been a long time since I tried it and I should devote a little time to seeing if blending a little grit in is a plus for my needs. But, I only use it as my DI, and don't really use/want it coloring the tone of the head to the cab. First world problems!
On my old Peavey head I used to distort one channel and blend it with the other for a little texture for Red Barchetta. For SS it wasn't a bad sound.


