Re: Prospecting New rivers
First there are a few questions that need to be answered.
Are they wild, holdover or frshly stocked fish?
Can you see fish actively feeding to get an idea on where to fish in the water column?
Is the angling pressure high?
What types of food are available to the fish.
These major things need to be addressed when standing on a river for the first time.
As for techniques, I like to nymph fish so that would be my primary approach.
However, if the trout were extremely active I might choose wet flies.
If the water is high and dirty I might streamer fish or nymph with attractor type patterns.
I would not be concerned with hopping from spot to spot as I would be focusing on fishing one area really well. This will tell you the type of water that the trout there prefer. This might aid you in finding fish in other sections. For example, If the river gets pounded you might fish the "good" spots and only get a few. However, if you fish secondary locations you might find lots of eager trout. Don't try to cherry pick when visiting new streams. Spend lots of time covering one area and you will learn a lot about the new waterway.
Re: Prospecting New rivers
When prospecting, don't neglect the small details. Watch where the mergansers go, there will be fish there. If you see an osprey or a heron take note. If you see a flash of a sucker on the bottom, you can bet there will be trout nearby. Sometimes the suckers are less wary and give their location up.
I good time to prospect a new stream would be right after a slight rain. The slightly discolored makes the fish a bit less wary and moves them out closer to the stream edges. It would be hard to beat prospecting with a small dark streamer in that situation. If you catch a few, you can always mark that location for nymphing under better hatch circumstances.
Don't forget to do a careful Google Maps search of any new area you want to prospect. In some of the better quality imagery, you can even see stream features such as riffles.