To get the exact beta on the Pucon creeks talk to the two Pucon oufitters above. From my own experience I have found that January is pretty optimal and after that the creeks start drying up. I usually try an schedule my Chile Kayak Safari trips (which includes the Rio Fuy) no later than the middle of February for this reason.
After the creeks start drying up there is still plenty of water left in the bigger rivers. The Futaleufu is huge in February and March. You are also likely to still find good amounts of water on the Rio Trancura that runs through Pucon.
Chile also tends to get a lot of rain. Usually the further south you go the wetter it gets. So you might luck-out with a rainy week here or there in the summer to boost water levels.
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Good Luck !