As a general rule, you can fly fish year-round. March through October are going to be more productive with longer windows of opportunity. November through January will be limited to the warmest part of the day with less productivity.
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One of the most difficult trout to catch because they are easily spooked and difficult to fool. Brown trout have more nocturnal habits. Target them with minnow-imitating plugs, spoons, spinners and streamers, various moth imitations (especially at night) and with cast or trolled live shiners.