Does a monk’s Unarmored Movement speed increase apply to the fly speed gained from a Fly spell?

A monk’s Unarmored Movement class feature increases the monks speed by 10 feet at level 2, and further at higher levels. According to this Sage Advice, this stacks with an Aarakocra monk’s fly speed, because that fly speed is innate to the race.

However, how does this interact with fly speeds gained from other sources, such as the Fly spell? That spell specifically states that “the target gains a flying speed of 60 feet for the duration.” Can this gained flying speed then be improved by a monk’s Unarmored Movement, giving a level 2 monk a flying speed of 70 feet with the Fly spell cast on them?