We are on a very tight budget (students) and *at least* 80% of our income after rent goes to food. Because of this we have faced precisely this problem!
Nowadays we mostly don't eat any meat besides lamb (not organic, but from a good source so not CAFO) and occasional organic beef (here that means it has eaten grass in the summer and grain/hay in the winter and the last few months of its life). Also occasional wild-caught fish. We don't eat chicken at all anymore because non-pastured chickens, even organic, are almost uniformly raised in horrific conditions and we can't afford the pastured ones. Apparently in the US the only difference between "organic" and "conventional" poultry is that the organic has had an organic diet. They still grow up and are slaughtered in appalling conditions. Of course I'm sure it depends on the individual farms, but I personally will not eat non-pastured poultry anymore because the great majority (like 99% in the US) of it comes from factory farms. As for pork, we mostly don't eat that either because we don't have access to pastured. Not only does that affect the quality too much for us to justify buying it, but we feel that pigs are so intelligent that raising them in any conditions but pasture is wrong.
Until quite recently we were getting slacker with our meat choices, mostly due to the budget and convenience. Then I read Eating Animals by Jonathan Safran Zoer. I would highly recommend it. The author is vegetarian himself but doesn't condemn meat eating. His book explores ethical and environmental issues around eating and raising meat animals, and devotes a large portion to describing factory farms and slaughterhouses. Absolutely horrifying in every way. He also describes small family farms that raise happy, healthy animals as well as small family-run slaughterhouses that kill meat animals more humanely, and then goes on to explain how factory farms are constantly running these small enterprises out of business. Not only that, but the "organic" label means very little for most meat, so if you can't buy it from a small farm/pastured it's probably factory farmed at least to some extent.
I suppose I knew all this already, but somehow being presented with it in such a clear and non-ideological way made me reconsider my food choices. I'm certainly going to continue eating meat, because it helps me be healthy and I don't think it's inherently wrong. But I think for my personal integrity I need to limit my choices to only what I know is not causing a mountain of social, ethical, and environmental problems, and unfortunately that weeds out a lot of the choices, particularly in the US (I live in Europe, where it's not quite as bad but still going in that direction). If it means that my budget can't include the variety and quantity of meat that I would like, I have to accept that, not just buy some of my meat from less good sources.
Anyway, I'm really not trying to moralize, just trying to explain what has gone into my own decisions in similar circumstances. If I were going to continue eating all kinds of meat but couldn't afford the best sources for all of them, I would definitely prioritize poultry and pork as those seem to have the worst problems in conventional production. Beef and lamb, even in the States, mostly spend a lot of their lives on pasture even when they are conventional (if I understand correctly) and just go to feedlots in the end of their lives to be fattened up on grain. Still bad, but not as bad as a pig that eats nothing but soy (even if it is organic) and never sees the sun.