On the very bottom left of the map window (in regular Google Maps, not My Maps), you should see a small ruler icon. Click on that and start dropping points along the river. You should be able to drag the map in between clicking points to make new sections of the river visible.
If I were you, though, I'd take the results with a hefty grain of salt, because a) the differences between the points you've dropped and the map's river outline, small as each one may be, will add up, and b) the river outline on the map may not accurately reflect the water course in real life.