If people in general would more often pick lower integers as private keys than larger integers, then it would be a good strategy for an attacker to start with lower integers.

However, the opposite also holds. If people would more often pick larger numbers, then atackers would ideally start at the end and work their way backwards.

In practice, no such bias exists. As long as people use decent proper random number generation, every integer is just as likely as any other, and it does not matter in what order an attacker tries things. He'll need to go through half of private key range on average regardless.

So no, there is no minimum number to start at. Yes, you have an extremely small chance of picking a number between 1 and 1000. But similarly you have an equal chance to pick a number between 76136428194729137 and 76136428194730136. An attacker has no reason to think you're more likely to pick something between 1 and 1000, so why would an attacker start at?

Also, whenever the attacker has the public key, there exist far faster (but still completely infeasible) algorithms to find the private key. These algorithms inherently operate in random order, so it does not matter.