Huh, interesting discovery today. When I turn my amp way up, my guitar sounds much more like I want it to, so I guess the tone is in there somewhere, just not with the amp quiet. That's kind of annoying...
Woah now, this kind of changes everything. So the tone you're after is available to you, but just not at an acceptable volume? Like Milnoque suggested above, a power attenuator is made for exactly this situation. He suggests the Weber, myself I use a THD HotPlate. There's a difference in price and features, and I've never used a Weber. I absolutely love my HotPlate, but a Weber might satisfy me just as well, who knows. Either way, being able to take the edge off of a cranked tube amp is almost a requirement for anything over, say, 15 watts if the speakers are very efficient at all. Before going through the trouble of selecting, installing, and possibly returning a new pickup(s), give a try to a power attenuator and see if you can get that cranked tone that you admittedly like, only at a volume more suitable for your situation. An attenuator is plug-and-play, and is much easier to return if you don't like it.
Now I don't know what amp you're using, or even if its a tube amp. Attenuators are ONLY for tube amps. They're not designed for solid-state amps. But I'll ASSume that you're playing through a tube amp.
One good thing about being able to crank the master volume and turning down your preamp gain/volume is that it gives you a lot of clean headroom. this allows you to have lots of "punch" and "snap" in your guitar tone, so maybe that'll get you in the neighborhood of what you like about a Strat.
Then, if you have a good overdrive pedal, you can kick that on and/or turn up your preamp gain and it'll give you a much more compressed tone; a meatier, thicker tone that'll really make a bridge humbucker sound nice. Maybe that'll get you into the neighborhood of what you like about a Gibson.
Either way, cranking the master volume on a tube amp is almost a prerequisite for taking advantage of what valve technology is reknowned for.
I'm learning every day how much I like having an EQ pedal in my signal chain. I have the MXR 10-band graphic EQ, and it can really change my amp's character pretty drastically. It's taken a few months of fiddling with it, but I've already found a few settings that really compliment a clean tone, a crunchy tone, and a heavier tone. Maybe that would be something that would help you get the tones that you're after from your current guitar and amp. Boosting or cutting the right frequencies can help you go from a Strat-like sparkle to a Gibson-like chunky darkness with a few slider adjustments.
Anyways, just another couple of suggestions to throw on the pile. Hopefully they help more than my previous suggestion of the Carvin guitars that turned out to not really be at all what I thought they were
