It almost works.
You're already moving horizontally because you started from the source of the Earth. If you did this at the equator, you'd be moving East 465 meters per second. By moving vertically, you'd increase your distance from the center of the Earth, effectively increasing your radius.
The larger the radius, the larger the circumference (C=2*pi*r), so the horizontal 465 meters per second you're moving would be a smaller portion of your circumference. So, when you came back down, the Earth would have rotated a greater percentage of its circumference than you did, meaning you'd land to the West of where you started.
In fact, thinking of it like that is stepping towards thinking of angles in terms of how many radiuses you've swept out around the circle (except the plural of radius is actually radians, not radiuses).
By the way, gravity is why us and the atmosphere stay on or near the surface of the Earth. The reason you don't feel the rotation is because the Earth rotates at a constant speed and carries you and everything around you, including the air, at the same constant speed - just like how you don't really feel how fast you're moving when driving a constant speed down the interstate.