By Anne Fisher
June 22, 2012

FORTUNE — Dear Annie: Our division of a huge global company is being sold (at a bargain-basement price) to one of our chief competitors, so we are getting a new boss whose mission will be to return us to profitability after several very bad years. We’ve already been decimated by four rounds of layoffs and we expect there are probably more to come. Meanwhile, my little team of about 20 people has performed well — we’re actually in the black, thanks largely to heroic efforts by my direct reports — but I wonder if that’s going to matter to the new CEO. I want to keep my own job, and my team’s too. I’d welcome any suggestions about how to do that. — In the Shadow of Doom

Dear Doom: It may well be true that more layoffs are coming. But, says Raymond Yu, there is another side to that coin, and it’s far less dark. “Getting rid of everybody and replacing them with all new people is just not practical, or even possible,” he notes. “So a new CEO is going to be looking for ways to turn the company around using the existing resources. He or she is going to look for the gems, meaning, the people who ‘get it.’”

Yu speaks from experience: He is CEO of Portland-based projector company InFocus, which at its peak in 2000 had about $900 million in revenues and 1,200 employees. Then a flood of low-cost Asian products slammed the company’s market share. After years of dwindling sales and negative earnings, investor John Hui took InFocus private in 2009 (also for a “bargain-basement” price, as you put it: $39 million) and, last year, put Yu in charge.

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InFocus is now in its third straight quarter of profitability, with 2012 sales expected to reach $200 million, a 30% jump from last year’s $160 million. Much of the growth comes from a product called Mondopad, a popular 55’ wall-mounted tablet for conference rooms.

InFocus is a far leaner company than in its initial heyday, Yu acknowledges, with only about 90 employees. “But many of the same managers are still here,” he says. How did he decide who stayed and who didn’t? “Someone trying to turn around a company is going to look for people who understand the business and its history thoroughly, but who are not so tied to the past that they can’t change,” says Yu. “The most important thing you can do is be open-minded and willing to do things differently. The last thing the new boss wants to hear is, ‘That’s not my job.’

“A business that’s been in trouble for several years running has fundamental problems that need radical solutions,” he adds. “You want to show by your actions and your attitude that you are open to those — or willing to suggest some. It’s an extremely stressful situation, but don’t let the stress paralyze you.”

Right now, before the new regime takes over, is the moment to put your team (and yourself) to work coming up with ideas for stemming the flow of red ink, Yu says. Even if your proposals end up on the cutting room floor, the effort will demonstrate that you want to help, not hinder, the new CEO’s agenda.

Kevin Coyne agrees. “The danger of being pushed out when the new CEO arrives is real — but so are the opportunities, if you quickly establish your value,” he says. A former senior partner at McKinsey, Coyne is now a professor at Emory University’s Goizueta School of Business in Atlanta and managing director of the Coyne Partnership, a consulting firm.

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Coyne did an exhaustive study several years ago of what happens to managers under a new CEO. The results, published in the Harvard Business Review, may give you pause: The involuntary turnover rate in the first year soared from 7% to 32%. Moreover, only about 15% of managers who were still employed kept their old jobs, got promoted, or made lateral moves. “Everybody else was gone,” says Coyne. “And scary as the statistics are, if we did the study again in this economy, they’d be even worse, because companies are running so much leaner now.”

When Coyne and his fellow researchers asked CEOs what made them keep the people who stayed on, the No. 1 answer was that those survivors “understood the extreme pressure the new CEO was under to fix things fast, and they stepped up and offered to help. It may sound like a no-brainer, but virtually no one ever does it.”

Two other tips: “Be on your A game in the first six to 12 months,” says Coyne. “Cancel your vacation, and don’t miss important meetings. If you’re not seen to be on board, you’re in the way.”

And second, keep in mind that it’s not about you. “The kiss of death is to say to a new CEO, ‘I was always underpaid and didn’t get the recognition I deserved under the old CEO, so now you can fix that,’ or anything else that is me-centered,” Coyne says. “If you don’t focus on the CEO’s turnaround efforts to the exclusion of everything else, you’re out.”

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Try not to get bogged down in negativity, and encourage your team to think positive too. “When revenues and profits were still falling after big layoffs, and nothing good seemed to be happening, people got very down about it,” Raymond Yu recalls. “I tried to balance that by talking about the future: ‘Here’s what we’re doing to make things better.'”

Then, he adds, “Once the company starts to recover and succeed, one of the great pleasures is seeing people thrive in new positions, and being able to reward them for that.” In other words, keep your eyes on the prize. Good luck.

Talkback: If you’ve worked at a company in turnaround, how did you deal with it? Leave a comment below.

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