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As the first book describes it, volunteering in districts other than 1-2 and 4 is highly unusual and usually done only to save a relative which got chosen.
I wonder why that is. There are especially two districts whose tributes stand a good chance:

1. District 7 in the Hunger Games Wiki is described as "District 7 provides lumber for the Capitol, and its people are known to be good with axes."




2. District 11 has a big population, most of which work in agriculture. Those are people who have a pretty rough life and who will not be lost in the wilderness (like the district 12 miners or the factory workers from district 8.)
   True, they are not trained in combat like the 'career' tributes, but it seems they still have a good chance to win. Thresh got close to the finish and even twelve year old Rue got pretty far in the 74th games.

So why is no one desperate enough to volunteer to escape the clutches of poverty?

Poverty doesn't make for good fighters.

The people in the outlying districts live on the brink of starvation. They're deliberately oppressed, kept poor and ignorant and most importantly weak. Poverty implies malnutrition, and malnourished people don't tend to fight well. For the same reason, armies (especially in poor countries) tend to be more well-fed and in better health than the majority of the population. But what army is there in Panem? The Peacekeepers come from the Capitol or - like Career Tributes - from District 2. From the authorities' point of view, there's no need to ensure any of the people in the outlying districts are in good health or properly fed, especially as children when they're not so useful as labour power.

Sure, some of the tributes from District 7 might be able to use axes, or those from District 11 might be good at living rough, but that usually doesn't make up for the years of combat training that the battle-hardened Career Tributes from Districts 1, 2, and 4 have been through. Career Tributes, brought up in relative luxury, have been able to afford the time to be trained, while those from poorer districts haven't.

Add to this the fact that the tributes are only children. Sure, District 7 has big strong men who can handle axes, but a boy or girl of fifteen won't be as strong and certainly not as efficient in personal combat. This, though, is something that puts District 12 at more of a disadvantage than the others: as Katniss notes in Catching Fire, kids in District 7 are put to work on the trees at an earlier age than those in District 12 are allowed to work in the mines. Which brings us to ...

Maybe there are more volunteers in other districts.

Canonically we know that volunteering is almost unknown in District 12:

District 12
hasn't had a volunteer in decades and the protocol
has become rusty. The rule is that once a tribute's
name has been pulled from the ball, another eligible
boy, if a boy's name has been read, or girl, if a girl's
name has been read, can step forward to take his or
her place. In some districts, in which winning the
reaping is such a great honor, people are eager to risk
their lives, the volunteering is complicated. But in
District 12, where the wordtribute is pretty much
synonymous with the word corpse, volunteers are all
but extinct.

_{-- The Hunger Games, Chapter 2}

But it sounds as though there could well be occasional volunteers from other districts, not just the Careers:

One by one, we see the other reapings, the names
called, the volunteers stepping forward or, more
often, not.

_{-- The Hunger Games, Chapter 3}

You're assuming that volunteering is almost unheard of in all districts outside of 1, 2, and 4, but I don't think the text of the book actually says that. It could be that it's most common in the districts of the Career Tributes, less common but still occasional in other districts, and rarest of all in District 12.

The tributes from districts 1 and 2 almost always win because they train all their lives (even though that's against the rules). Almost every year they see someone from district 1 or 2 win. It's obviously all but a death sentence, and given the relative poverty in the outlying districts it seems like nobody there will have the time to train since birth.

Maybe because people have the belief that starvation is a lesser evil then the cruelties of the games,
Just because your starving to death and need food doesn't necessarily mean you want to be killed another way like tracker jackers turning you insane or mist that burns you slowly to death or mutts that slowly eat you alive or drowning or the countless other ways you can die in the games, maybe most people have the belief that they'd rather just starve then be killed what they believe to be a more traumatic way.

With the exception of Districts 1, 2, and 4, the other districts (especially the higher numbered ones, such as 8, 10, 11, and 12) live in abject, dire poverty. Poverty and undernourishment makes a person weak, and therefore in poor shape to fight, and therefore be able to win the games. By contrast, tributes from wealthier Districts (the Career Districts) are better fed, and have opportunities to train for the games from a young age. This is why they almost always place highly in the Hunger Games.