Telekinetic only extends the range of mage hand
The Telekinetic feat grants you three independent abilities:
- Increase your Intelligence, Wisdom, or Charisma score (...)
- You learn the mage hand cantrip. (...) If you already know this spell, its range increases by 30 feet (...)
- As a bonus action, you can try to telekinetically shove one creature you can see within 30 feet of
you. (...)
The ability to shove is a separate, independent ability from the mage hand ability and its extended range. It does not have anything to do with mage hand, and it is explicitly limited to 30 feet. This ability does not state that it extends to 60 feet when you already have mage hand before you take the feat, so it won't.
You can see that the individual bullet points of a feat are independent from each other unless they explicitly state otherwise from this answer on Crossbow Expert in the Sage Advice Compendium:
Is it intentional that the second benefit of Crossbow Expert helps ranged spell attacks? Yes, it’s intentional. (...) The second benefit of Crossbow Expert prevents you from suffering (...) disadvantage, whether or not the ranged attack is with a crossbow.
As for the related question, the range of a spell is a special attribute of the spell, distinct from other distances mentioned in the spell's text. Telekinetic interacts only with the range, not with any other distances mentioned. This is explained in detail in this excellent answer by Thomas.
If you already have mage hand, allowing you to also extend the distance for shove or other distances mentioned within the text of mage hand would likely not be overpowered from a game balance perspective. The rules are there to serve you to have fun, so if both you and your DM agree this makes more sense for the feat, you are free to make a ruling and change this.
