It depends on your plans.

Normally a plan will not allow a loan to be taken from a terminated employee, so it would not be likely that you can take a loan from your prior employer plan.

For your new plan, you would be eligible to take a loan as soon as you have a vested balance that would qualify you for the minimum loan amount. For example, if new plan allows loans in a minimum amount of $1000, you would need a $2000 vested balance to be able to take this $1000 loan. As you noted, depending on your contribution level it may take some time to be eligible for that loan.

There is another alternative. Since you mentioned you have an old plan with money in it, you could rollover that balance from the old plan to the new plan. That will cost you nothing but will consolidate all of your retirement plan money into a single plan. Then, you could take a loan from that money in the new plan. Depending on your vested balance in the old plan this may be a much faster way to take a loan of the size you need. To do that, contact your old plan to initiate the rollover.

The best idea is not to take a loan from a retirement plan, but if you must, these would be your different options.