Should I send a second reminder for recommendation letter?
If this is for an application for a US graduate program, the due dates for the main application are stricter than for recommendations. The admission or review committee likely won't meet until next year. I had an advisor that had a habit of submitting these things a few days after they were supposed to be submitted and it was never a problem. Since they acknowledged your reminder email, I suggest you leave them alone. If you arrive at the due date with no submission, then you can send another polite reminder if you feel like its necessary.