Question on using JR Pass

What I'm wondering is if JR passes can be used to pay part of the fare for a full trip that has sections outside the area covered by the pass. I'm not sure if I'm phrasing that well, so let me explain.

I'm starting in Kamigori and ending in Nagoya. The JR Kansai Area pass seems to go as far as Maibara on the route to Nagoya from Nagoya. What I'm wondering is: Can I go all the way to Nagoya Station, show the pass there and then pay the difference in fare. Or do I have to get off at Maibara, exit through the gates and then purchase a ticket to Nagoya.

You can travel to Nagoya and pay the fare difference at the exit gate.
Pls note that the Kansai area pass does not allow you to ride on Ltd Express trains with the exception of the Haruka. You need to change trains a couple of times, and the average time taken is 271 minutes.

This spring I had the Japan railpass and I went to Amanohasidate. Part of the trip is not valid with the pass. I simply buy the ticket directly at the conductor with no Problem.
I think it will be the same.

Iirc if you buy the ticket in advance (I book all my trains in advance, including express trains) at the midori no madoguchi, they'll just charge you the difference and give you a ticket.

I'm currently in Japan with a JR pass. I've already got a reserved seat for Tottori to Himeji which is mainly on private track. All I do is pay the extra either on the train or at the ending station.

This is also the same for most other trains. I'm not sure about the Meitetsu/Kintetsu lines though as they are on their own system...

Or, as MilesVor says above, you can just pay the extra fare before travelling.

The last but one time we were in Japan, we did exactly that while travelling from Kanazawa to Echigo Yuzawa with JR Passes.