Definitely don't make any after the lender has requested the payoff amount, as that will really mess things up.
If you are going to make a payment before the payoff is ordered, here is my opinion.
Payments you make will have a higher interest rate, but will lower the new starting loan balance on the new loan (slightly). So if having a lower loan balance (which would result in maybe a dollar or two a month lower payment) is important or if you are short of the 20% (or 25% or whatever) equity, then pay as much as you can now. If not, only pay what you are required to pay, and wait for the lower interest rate. You'd save a tiny amount in interest by paying early, but unless the difference in interest rates is HUGE, I don't think it would be enough to make up for it.
Note: I didn't do any
math to come up with this decision, not knowing most of your numbers, but logically, it seems right.
Also keep in mind that if you are refinancing, usually the prorations (the prorated final to the old lender and the prorated first to the new lender) end up either in the payoff, or in the closing costs, so there is often one month that you don't make a regular payment anyway. So what you are calling "October's payment in advance", may end up being applied as a principal payment, and you will still owe for October anyway, as part of your closing costs. So whatever you are trying to accomplish by paying early may backfire and you may end up having to pay the equivalent of 13 payments this year, instead of only 12.
The best thing to do is to talk to the lender doing the refi and ask them how prepayments affect your numbers.