Steelhead. Steelhead should be well-spread out between Sherars Falls and Warm Springs, and there will still be a few fish between Sherars and the mouth. Some anglers feel that by this time of year the fish are more receptive to drab patterns and smaller sized flies--probably because the fish are seeing lots of anglers.
Proper line choice--sink-tip or floater--is very important this month. The choice is not always simple. See Sink-Tip or Floating Line? for some advice on the subject.
Something that a lot of anglers ignore is salmon spawn. This can be very important between Sherars Falls and the mouth. Trout (and sometimes steelhead) take spilled roe from spawning salmon. So look for areas with salmon, then cast an Egg Fly below it. Take special care not to disturb the salmon or walk on their redds (nests; distinguishable by the clean-scraped gravel).
Trout. Trout fishing should remain good for most of the month. Some green caddis (Rhyacophila) and saddle-case caddis (Glossosoma) are still around in October; mostly the latter. Size 14-18 Elk Hair Caddis with either tan or olive bodies are good fly choices; see Jeff Morgan's CDC Everything for a good pattern. Soft Hackles or other emerging/egg-laying imitations in sizes 14-18 should be in your fly box, too; use flies with brown or gray bodies.
Blue-winged olive hatches can be heavy at times. Two types of blue-winged olives are present: size 18 and size 20 (the latter is the genus Acentrella). Be prepared for both and carry some olive size-18 and size-20 Sparkle Duns. Similar sized Pheasant Tails are always a good choice at this time of year.
October caddis are still present, so fishing the banks with an orange-brown size 8 Stimulator or Madam X with a brown-orange body is a good strategy. Some anglers match this hatch with a bright orange imitation, such as they'd use during the salmonfly hatch. I find that fly too bright for the October caddis and prefer a more subdued body and a darker wing.
When casting October caddis dry flies, look for water that is 2-4 feet deep and flows at a moderate pace; a rocky bottom is a real plus. Sometimes a dead-drifted dry fly works well, especially on cloudy days. If the trout are ignoring you, however, try skating your October caddis imitation across the surface.
Another way to pick up trout during the October caddis season is to tie on a size-8 Soft Hackle or Serendipity with a creamy-orange body; let it tumble near the bottom. Local fly shops carry good October caddis pupa patterns.
Midge hatches will be a frequent evening occurrence this month. Carry a seine so you can check the size and color of whatever is drifting down the river. Midge fishing is best in quiet runs, in backeddies, and near rocky banks that create mini-eddies. You'll find whitefish feeding in the slackwater areas, and trout where there's more current; sometimes only a couple of feet will separate the two species of fish. A midge pupa pattern is usually the best choice during a midge hatch, but a Griffiths Gnat can also work well.
A few mahogany duns, aka "paraleps" (Paraleptophlebia) are still hatching, but they should taper off by mid-month. A size 14-16 Sparkle Dun with a dark brown body is a good choice when they're hatching.
If you're having trouble picking up trout on dries, don't hesitate to go subsurface. It's a good month to cast a two-nymph rig with a big stonefly, such as Rubber Legs or Kaufmanns Stonefly, on the point and a size 18 Hares Ear or Pheasant Tail on a dropper or trailer. Cast into slightly riffly water or runs of moderate speed.
This time of year you will find more trout in backeddies and in slower water near the bank than you will in faster water. The best trout fishing should be in the late afternoons and evenings.
Keep in mind that it's October. Expect cool days and sometimes bitter winds. Bring extra clothes in case you slip and get wet. If you're headed over here from the Willamette Valley, be careful on the roads. I had a narrow escape a few years ago when an isolated shower of freezing rain coated Hwy 26 with ice. I hit the ice at 60 mph while towing my boat and left the road. It's not an experience I wish to have again, and I hope you avoid it too.
If the river suddenly rises, such as following an intense rainstorm, fishing will be poor until the extra water subsides or stabilizes at a consistent level for a few days.
White River can still be ill-behaved during October. A major rainstorm on Mt. Hood can cause major murkiness below this river's confluence with the Deschutes. If you're concerned, call a local shop such as Deschutes Angler Fly Shop or Deschutes Canyon Fly Shop.
For more on October tactics and flies, see the Rivers in General forecast