I'm set up with a hot HB and a low output single in my neck. I love being able to clean up with the volume knob even when a drive pedal is on. But I also want to get that juicy single coil lead sound for legato and so I need a 2nd gear. I'm looking at maybe a control switch or expression pedal. Anyone else have low output singles that they juice or should I choose the right tool for the job...as in a different guitar with a hotter neck p/u?
Your over all description of the desired tone sounds like you want a more compressed tone, not necessarily more gain. I think of these things in their simplest, more traditional ways. So my inclination is to suggest compression first. Play around with not only the compression itself, but where you place it in the chain: anywhere from the first block in your chain to the last block before your output.
Once you get compression where you want it, if more gain is desired an X/Y switch on your overdrive could help you switch back and forth easily (and can easily be set up in scenes) between standard overdrive and a slight boost for solos. If you prefer to X/Y the actual amp for different overdrives that's an option too, but it doesn't change the approach.
Because you have a low output pickup, a slight boost in overdrive could just give you a satisfactory solo boost, but overdrive would need to be located after the compression, otherwise the compressor would no doubt nullify any volume boost. The tone would change. The volume would likely remain the same.
To remedy this, I would add an EQ block to the scene and boost mids slightly. Not only would this help you pop out in the mix for solos, depending on which frequencies you boost it could easily accentuate the juicy legato tone you're after.
All of my suggestions could easily be set up as scenes for simple toggling back and forth.
As this thread implies, there are many ways to handle this. The above is how my brain works.
Good luck!