Good advice so far. Platte river special is a must on the mile,or the platte river in general. This time of the year you will do better with midges, caddies, and pdms.
That being said streamers will work on cloudy days, and dry flies are another option at this time of the year. Don't forget about hopper patterns. Check out the north platte lodge for the scoop on what is the hot ticket for the Mile, the reef, and every thing in the middle.