The Ceriatone Horsebreaker has an huge range of tones. It has a BluesBreaker and a Klon side. You can stack ’em, use them individually and there’s a switch that lets you decide which side comes first. And it’s reasonably priced for a handmade pedal.
Get a dual OD, any of the big ones work well with the Mesa Cleans. I like the Gladio, between the two sides you can get a lot of different edge of breakup to dirty sounds. The Protein is getting a lot of love these days as well.
My suggestion for indie stuff like The War on Drugs and MM is to get a RAT and set it to mild distortion. Then you can roll off the volume knob and get it to clean up into a nice, crispy overdrive. That's two pretty quintessential indie rock sounds. Though I don't know that it'll scratch the blues itch, so might not be the best all around option for you.
FWIW, I’ve found the following OD’s work really well with my Mesa Boogie Fillmore 50:
Shnobel Tone Daily Driver
Barber LTD SR
Barber Gain Changer SR
Skreddy Screw Driver Mini Deluxe
BearFoot FX Honey Bee X
SolidGoldFX Zeta MkII
And please know… this isn’t just a list of my favorite OD’s, but in fact I’ve found them to work well with my Fillmore (different amp from your California Tweed but similar genetics) and also good for indie/rock/Stones/blues-type OD tones and flavors. Most of these suggestions are also pretty versatile.