Supplement fare for Odoriko to Izukogen?

I'm traveling to Japan for the first time (leaving tomorrow: 12/11!) and part of my trip will take me to Izu. I've just read I will need to pay a supplemental fee for the track beyond Ito. I want to take the Ltd. Express Superview Odoriko 3 from Shinjuku to Izukogen. I have a JR Pass. Do I need to pay an extra fee for this trip? If so, how do I pay? Do I buy a ticket or pay the fee at a ticket booth while showing my JR Pass? Thank you for any help.

Do I need to pay an extra fee for this trip?

Yes, the section beyond Ito is not covered by the Japan Rail Pass. The supplement fare is 640 yen for the base fare, 400 yen for the limited express fee and 100 yen for an eventual reserved seat.

If so, how do I pay?

If you travel by unreserved seat, you can pay the supplement to the conductor on board of the train. Or, if that does not happen, at the manned gate of Izu-Kogen Station.

If you want to travel by reserved seat all the way to Izu-Kogen, it becomes more complicated, because part of the seat reservation is covered and part is not. I am not even sure whether the staff at the JR counter is able to sell tickets for just the non-JR section. So, I am not exactly sure how exactly this would be dealt with. You will most likely get two sets of tickets and seat reservations; one from Tokyo to Ito and one from Ito to Izu-Kogen, and pay just for the latter.

Thank you so much for your quick response!

I guess I'll try to do the unreserved seats -- seems more straightforward. I'll make sure to have cash for the fee.

Thanks again!