The key to managing panic levels is building satellites and satellite uplinks. When you place a satellite over a country, it lowers the panic level by 2. The more coverage you have over a continent, the lower the levels across the entire continent. Also, it seems to keep panic from rising as fast as it does when there isn't a satellite, but not sure on that point.
Satellites also add the extra income and engineers you need which makes it easier to build/buy other things later.
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Building a Satellite gives an immediate -2 to panic level in that location, with a chance at further reduction at the next Council Report. Completing Terror Missions with a high enough rating reduces panic across the entire region, not just the specific nation the mission occurred in, and some story missions reduce panic across whole continents, or even worldwide.
France has decided it will surrender to the aliens first in my game. Started in easy mode and the game jumped straight to very difficult missions before the end of month 1, and showed me a base in month 2. I haven't visited the base yet, as I'm completely broke.
Countries won't leave immediately, they wait until the monthly report. Save your satellites until the day before the meeting, then deploy according to panic. I've been close several times to losing nations, and through monthly dedication to uplinks and satellites as well as holding off until the day before the council report I've managed to keep all of them so far. The idea is that if you deploy early in the month it lowers panic, but you still have all those days to go in which abductions can raise the panic again. Since a country in the red won't immediately leave, it's best to leave them where they are and take care of it at the end of the month.
Besides the listed ways, the only other thing you can do is to assist the countries with the highest panic levels when given the option. You might lose out on better rewards, but at least the nation won't leave XCOM anytime soon.
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