One off my pedals is always on by design, a preamp/buffer/boost by Pettyjohn Electronics called the Lift. But I don't think of it as a pedal, I think of it more as a conditioner and preliminary EQ (I like to cut a little low frequency content just a touch to prevent muddiness in recordings).
However, they also made a pedal called the Edge that's nearly always on when I use most of my amps. One of their pedals called the Chime is on most of the time for the DG30 amp. Both of these are low gain pedals. I use very little dirt when I play through them. I don't use a dirt pedal with the HXDA, that amp needs nothing.
The Edge is the only 'grit' pedal I find useful with my Mesa amps; for some reason they seem to dislike dirt pedals, even clean. But the Edge works well with them.
Unfortunately, Pettyjohn is currently only doing limited runs, but my goodness, their pedals sound great because they use studio-grade electronic parts. Here's the Edge demo I think really does justice to the pedal. You have to click on it because it's Vimeo, and you can only watch it on that site. Still, it's great.
I don't know if it's still possible to by an Edge at retail. Dealers might still have some hanging around.
I think of my Pettyjohn stuff as 'betterizers', however I have one pedal from Suhr that I use as an always-on for very clean tones with a couple of amps, called the Kokoboost. I don't even use it as a boost for the most part, I set it to unity gain, so the volume doesn't change, but for some mad scientist reason the pedal makes the signal hitting the amp sound better. It's subtle, but it's there.
Anyway, that's what I use. I did try the Morning Glory, and it wasn't my style, so I returned it. I mean, we all have our preferences and all are legit, it's whatever works with your rig, and that's going to vary from player to player.
Here‘s my board. The Edge is lower right, KokoBoost is upper right, Lift and Chime lower middle. Signal hits the Lift first (to the left of the expression pedal), then >> Kokoboost >> Edge >> Filter (EQ Pedal) >> Suhr compressor >> Suhr tremolo > Suhr chorus >> tuner >> H9 >> patch bay on side of the board so I can use the second H9 separately with a second amp, or at the end of the chain playing through one amp (the patch bay is normalled with the second H9 in the chain, but the normalled signal chain can be tapped between the two H9s to use one with a second amp, or to insert external gear like a Tape Echo, volume pedal, etc).