I used an online bill payment system to schedule electronic payments for one day prior to the payment date. Normally, it worked great.
Then, I had a fraud alert on my checking account and it was suddenly closed, and the funds were moved to an alternate account. Unfortunately, the web-based bill payment system didn't sync up with that change. And my mortgage payment was scheduled for the next day. And it promptly bounced, causing my lender to asses a load of additional interest, fees, and penalties.
I now schedule my mortgage payments for at least a week in advance. ;-)
So my advice: if you're not having cash flow issues, and you're not worried about the small change the additional savings interest will provide, I'd say just go ahead and pay in full as soon as you get the bill. You might save yourself the headache of calling a lender and pleading for them to back out fees, etc.
When it comes down to it, my time is worth more than the money I'd save being pedantic about due dates. I value my time with my family more than the time I'd spend on the phone sorting out late fees. Your situation might be different, so that's your call.