That, my friend, is the age old dilemma isn't it? Switch from Strat to Les Paul and it's muddy. Switch from Les Paul to Strat and it's weak and thin. The most common solution I'd say is have two overdrives, one for your humbuckers and one for single coils.
When I used to switch between my Les Paul and vintage output Strat a lot I had two different solutions.
The first was that I used a true bypass loop switcher, it had 6 loops all together; Loop 1 had a wah and POG, Loop 2 was a compressor that was set up for my Strat, Loop 3 was a comp that was set up for my Les Paul, Loop 4 had two overdrives that were dialed for my Strat, Loop 5 was overdrives dialed for my LP, and Loop 6 was time based effects (delay, reverb, etc). It made for a HUGE pedalboard that was totally overkill, but it made it easy to switch between "presets" without tap dancing. That way I could have my amp dialed in and never have to adjust it, and if I adjusted a pedal for my Strat, it didn't affect how my Les Paul sounded.
The second solution, and my favorite one, was a Seymour Duncan Pickup Booster. It would be off when I was using the LP, and then switch it on when using the Strat.
It did two things for me; 1) Boost guitar signal to maintain the same input level into the amp so there wasn't a volume drop or significant change in gain/compression from my pedals or amp, since I was using low output vintage style pickups, and 2) It had a switch that beefs up single coils by shifting the resonant peak frequency down, so I wouldn't go from thick/rich humbucker tone to weak/thin single coil, basically just gave the strat a little more oomph in the mids and less ice picky highs, while still having that nice strat tone.
Nowadays I use a Fractal Audio Axe FX and I have presets that I've dialed in for specific guitars. But I also only own one guitar right now and it's a Les Paul so I have no issues LOL