Stack size relative to starting stack size is less important than stack size relative to the current blinds. Just because you lose half your chips in the first levels, you don't have to panic. When your stack starts getting to be less than 10 BB, then pushing marginal hands becomes more reasonable.

If you have to call a preflop raise for a third or more of your remaining shortstack, and you have a hand you don't mind racing with (decent PP, AK/AQ), a push early is reasonable.

But there's no reason to be open-pushing 500 chips into a pot when the blinds are at 10/20 or 15/30 or something along those lines. You can still fold the flop and have 10-20 BB left with a standard 3/4x BB raise at those levels.