No. Haste doubles your speed, not special movement speeds. This is a bit confusing and one of the not-natural-language things, bur reread the movement section of the rules. "Speed" is specifically walking. Flying is a special movement form.
Except if something swims, flies, or climbs normally, Haste affects all of those as well.
While you might think of Fly as a special movement form, it isn't included in the PHB under Special Types of Movement, which only covers Climbing, Swimming, Crawling, and Jumping.
A better argument
against Haste affecting Fly is this:
General Rule: Haste doubles speed of the target. The target's speed could be walking, swimming, etc.
Specific Rule: Fly grants you a Flying Speed of 60.
However, this is all interpretation of the rules, and a table could go either direction IMO. Personally, I would be fine with either ruling.
Consider something like a Pegasus, with a Flying speed 90. Haste would make that 180, which is pretty fast (about 41 mph with move + dash).