Early September is usually an extension of August, with heat and low water making fishing difficult. But, you can't change the orbit of the earth or the tilt of its axis, so this month the sun is going to get lower in the sky, and the nights will grow longer. Even if we see high peak temperatures, it won't be hot for long. We're more likely to see a cold snap than a heat wave. If that happens, it will take trout several days to adjust to the cooler water. Cool, damp weather could muddy some streams, especially those whose headwaters are in Yellowstone Park.
You can expect the hopper/dropper/trico/caddis strategy--the mainstay of August--to last a through the first week or two of September. When colder weather hits, with nightime temperatures below the frost point, the hoppers will fade. They'll be replaced with hatches of size-18 blue-winged olives, mahogany duns, size 20-22 tiny olives (late in the month); standard nymphs, and streamers will work all month. Unless you're adamant about casting a dry fly, this should be a good month, all month. If hot dry weather comes back, terrestrials and caddis will return to center stage.
As mentioned before, the major hatches this month are blue-winged olives and mahogany duns. The mahogany duns (Paraleptophelbia) create trout feeding activity in quiet bankwater because that's where the nymphs migrate before hatching. If you see trout languidly rising in slow water near the riverbank this month, you're probably witnessing a mahogany dun hatch. DON'T cast blindly. In this quiet water you'll spook the fish. Instead, watch the rises and pick a single trout. Use a downstream presentation so the fly reaches the fish before the leader and line.
Blue-winged olive hatches will be sporadic, but nymphs are active and are taken by trout more regularly than the duns, especially in riffly water; see A Little Known Two-Fer-One for tips on how to take advantage of this. Cloudy weather will improve this hatch.
October caddis may show up by the end of the month. Trout take pupa patterns as well as adults. When fishing an adult October caddis pattern, you'll probably catch more trout by skating the fly across the surface than if you dead-drifted it.
And, although they are diminished in importance on most streams, tricos and hoppers will remain on some trout menus until the first hard frost. Caddis can be important, too; most will be size 14-18.