Does the "Sandstorm in a Balloon" have the improved double jump?

Yes, a Sandstorm in a Balloon has an improved double jump.

Reddit user DataNalle posted an infographic showing the jump heights of all jump-modifying items. As seen in the picture, the Sandstorm in a Bottle only goes up 20 tiles, while the Sandstorm in a Balloon boasts a distance of 36 tiles.

Note that the Cloud in a Bottle only goes up 11 tiles, which its in a Balloon variant only goes up to 19 tiles. So it's very clear that the Sandstorm in a Balloon travels much further up than a Cloud in a Balloon.

Also consider the fact that a Red Balloon increases your jump height by 10 tiles. This means you actually lose some of the effect by having a Cloud in the Balloon, since the increase is only 8, meaning you lose 2 tiles of jump. However, the Sandstorm goes from 20 to 36, so you get an extra 6 tiles of jump height when you upgrade.

So statistics wise:

• Sandstorm upgrade from Bottle -> Balloon (20 - 36) = 80% increase in jump height
• Cloud upgrade from Bottle -> Balloon (11 - 19) = 72% increase in jump height

So it's very clear that the Sandstorm variants offer a great increase in jump height over the Cloud variants.

As mentioned in the Terraria wiki, the Sandstorm in a Balloon keeps the "twirl" effect. In general, the respective effects stay the same, be it Cloud, Blizzard or Sandstorm.

The Sandstorm in a Bottle is the Strongest out of the 3 _____s in a bottle, which are the Cloud in a Bottle (weakest), Blizzard in a Bottle (medium), and Sandstorm in a Bottle (strongest). If you make the Bundle of Balloons, this difference becomes apparent. So yes, The Sandstorm in a Bottle does have an improved double jump.