Why search for a nonce linearly (+1) vs searching only even or odd nonces?

Why search for a nonce linearly (+1) vs searching only even or odd nonces?

As a solo miner, you have the disadvantage of a minuscule hash rate compared to mining pools. Is it not possible to searching for only even nonces? Of course you risk the chance of missing the nonce, but you gain an effective 2x hash rate.

I wonder how this or other nonce searching methods might impacts your chance on average.

  • What if you search downwards for a nonce?
  • Skipping (even, odd, every third, etc).
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