I used to encourage our bass player to use some distortion on his sound during my solos, kinda worked.

+1 ... and maybe adjust your EQ with more on mid-range

I spent most of my younger years doing the 3 piece thing and what I wouldn't have given to have an HD500 back then. One of the things to be sensitive about is that your solo level shouldn't have too much of a boost compared to your rhythm level. It's kind of a less is more kind of thing.  When you're playing with another guitar player you need a boost in level so that you can hear the articulation of the solo but if you use that same boost with just a 3 piece band you're going to find that you'll be way out on top of the bass and drums instead of having a nice mix.  When it's just bass and drums behind you you're guitar has no competition for it's frequency range the way it does if there is still a rhythm gutiar playing.  So I guess bottom line is that it's not about getting a bigger solo sound it's about getting a bigger band sound while you're soloing and that's why you think it sounds fine when you're playing rhythm and all the levels are even.

On top of that there are some other good suggestions regarding the tone itself.  Nothing too bright, I like to use the classic distortion and turn the filter up to 100% which really takes the edge off, kind of gets you part way to that neck pickup solo sound while you are on the bridge pickup except that you still get a bit more bit from being on the bridge pickup.  But really, the first thing to try is to bring your solo levels down.  If you listen to RUSH you can still hear all the notes that Geddy is playing in the background while Alex is soloing.  I would suggest all the years they have been around and doing it, they know a thing or two about how to do 3 piece bands.