That's a good question, actually. It depends on the pass. For example, for a JR Kyushu Pass, you can actually use it up to Shimonoseki in Yamaguchi (which is not Kyushu). So if you have another regional pass, you can just keep on riding. But in the example you mention there is no overlap, so it looks like you'd need to pay the fare for the extra ticket as you mention. And this of course assumes you have an activated pass. If you just have the exchange order, you'd need to get the Pass from it, which can be done in Hakodate. You can buy the voucher for the Hokkaido Rail Pass in Narita or at a JR East office in Shinjuku, but you can't exchange it for the actual Pass there.
So if you ride between Aomori and Hakodate, the section between Nakaoguni and Hakodate is covered, but you'd need to pay the fare for between Aomori and Nakaoguni.