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Once you have reached Nordkapp, your best bet to hitch out (which involves you return back from where you came) is to hitch a ride at the NordKapp Hall itself. You should at least manage to get a ride to the nearest town of Honningsvåg or with luck even manage to get out of Magerøya Island.
Anyhow, the distances are so big that walking out of the place is basically useless since Nordkapp is anyway the emission zone of all traffic in the area. The same applies when it comes to leave the island while hitching from Honningsvåg, it is better to stay at the crossroad of the E69 at the entrance of the town instead of walking along the road and into any of the 4km+ tunnels.
There is two things to bear in mind, the Tunnel to Magerøya Island is toll per passenger in the car (both ways). You might therefore have to pay your due. However, there is a general tolerance to tourists cycling, walking, and backpacking. To avoid paying the one-way fee (NOK 48,-/€6 per passenger in 2007), on the way to the island ask your driver to drop you out between the tunnel and the Bomstasjon (pronounced boumstashiôn), and to please wait for you on the other side. Just walk around the station. The people will not mind, as they have to charge passengers in the car at the tollbooth. You can wave at them to show your gratitude. On the way out of the island towards the mainland, walk around the station and hitch a few meters before the tunnel.
After the Tunnel, the only worth ride to catch is at least until the crossroad of the E6 to a place called Olderfjors (also often mapped as Russenes). However, if the locals offer you a ride to somewhere not that far, do accept it. You will be sure to learn a lot of local knowledge, might get to visit a genuine Finnmark cabin (hytte) somewhere along the road. Remember this is the only road, therefore you will not miss out on the traffic, as long as you are on a stretch of the road where you can be seen and where cars can stop.
At Olderfjord, try and talk to as many people as possible, as this place is little more than a crossroad. The E69 takes you to Nordkapp. The E6 brings you on one side to Hammerfest, Alta, Tromsö and the rest of the country. On the other side it will take you to Kirkenes (very few people driving that distance), but also to Lakselv, Karasjok, and Finland.