Clear Creek Route (east side of the mountain) is a good route for that time. There probably won't be any snow at that time or not much anyways. It's the least Alpine Experience route on the mountain. It's 3 miles from the trail head to Clear Creek Springs. Clear Creek Springs is the only water source on this route and where most people base camp. From here you climb up a broad back ridge around 20 degrees (lots of scree) until you get to the crux of the climb at around 13,000 ft. Here is where you cross the top of the Wintun Glacier. Depending on when you go, will determine if you will cross snow or not. It's always recommend to still have ice-axe, crampons and helmet even when there's no snow for the just in case (getting off route). Here's a bit of trivia for that route. It's the route that the first mules went up in 1883. Dynamite and Cropper. In 1903, the first horse "Old Jump Up" made it to the top. Unfortunately, he died the next day of over exhaustion. Poor horse.
For more information, stop by the Mt. Shasta Ranger Station. They have an excellent powerpoint of this route and all main routes on Mt. Shasta.
Cheers and Happy Climbing!http://www.mountshastaguide.com