I'd highly recommend the EBS Multidrive for a subtle, slightly pushed amp type of OD. They're not very expensive, especially if you pick one up used, well made and simple, with a good tone. Just gain and volume, tons of volume boost on tap if that's your thing, and a few modes to pick from. I like the "STD" mode for mild OD most of the time, the "TUBESIM" mode is a little bassier and seems like it has some clean low pass in it. I have a vintage RAT (paired with a clean blend) and an OCD v4 and the EBS Multidrive is least "distortiony", as you put it. I like the other for a more aggressive overdrive/distortion, but the EBS is right in the sweet spot for mild to medium overdrive along the lines of a slightly pushed tube amp to my ear. There are a few other pedals I've wanted to try that have been described similarly, like the Fairfield Barbershop, but they're usually more expensive and have a bigger footprint as well. Little things like active/passive switch and the small size push the EBS way up the list of very useful low key overdrives for me. It's not the best thing for every OD usage, but I think it fits the bill for what you're describing, in my experience.
I kid, I kid