Does the Telekinetic feat also increase the distance you can move the mage hand?

The Telekinetic feat from Tasha’s Cauldron of Everything says the following:

You learn the mage hand cantrip…. if you already know this spell, it’s range increase by 30 feet when you cast it.

Mage Hand has a range of 30 feet, and says the following:

The hand vanishes if it is ever more than 30 feet away from you or if you cast this spell again.

Does this mean that even though the range for Mage Hand is now 60 feet, if you cast it beyond 30 feet, it still disappears?
Also, Mage Hand says the following:

You can move the hand up to 30 feet each time you use it.

Can you still only move it 30 feet? Or can you move it 60?