Hey there Pelican, I had the same question a month ago haha and I just came back from NZ. I am a 1st yr regional FO so my options were to get to NZ via the cheapest route since working of $22/Hr is almost inhumane and I wanted to visit my grandma in NZ after 5 years of going to school and CFI'ing to get a regional job without any vacations.
My airline does not have any agreements for Air NZ so a direct flight out of SFO or LAX was out of the question. After some hard research I decided to go with HAL from HNL-AKL and back.

1.Now the trick is to call 1-8555-325-8948. Tell them your a pilot and want to list for the jumpseat. I had a hard time because I think the call center had a lot of foreign agents so they had a hard time understanding, but eventually they will get you listed and give you a PNR. I just created a listing for the return legs back from AKL -HNL. You will pay the taxes and fees in HNL (long story).

2. The next hurdle is in HNL. For jumpseating with HAL you have to go to the customer service counter. Tell the agent you would like to list for the JS for the AKL flight. I am 100% sure they will say that you cannot do it because of the fees/taxes and they will say it has to be done via myidtravel.com DON'T Believe Them. Many agents have never had anyone list for a JS for AKL and it can be done. I had to go on the alpa JS website and show her that as a regional ALPA approved pilot, HAL invites anyone. The supervisor came in and told me the same thing but I told her that our Pilot Union Contract and HAL pilots group say that I am allowed. The thing is they have to give a ring to "THE CALL AUDIT Room" phone number. They have a hotline trouble shoot number if they get stuck and thats what the supervisor did for mine. After she go through with someone at HAL customer agent help hotline she quickly processed my JS. They will ask you if you created a listing for the return flight, this is where you give them the PNR from the 1-888-number.

3. The agent then should be able to list you in the HNL-AKL flight and link it to the AKL-HNL return flight and process everything and create a total cost for your flight. My total cost was $59.xx roundtrip from HNL-AKL-HNL I paid by credit card with the agent. It is all etkted so it should be no problem on the way back at all. Your allowed 2 bags for free to checkin if you need to take anything. Ask her for the HAL 110 JS form, just to fill it out for the captain. You get a STBY boarding card.

4. Went through KCM, was in uniform, got through, changed into regular clothes. Went to the gate, gave the gate agent the JS form and STBY Boarding card. She then cleared all the nonrevs as boarding got close. got my seat, a nice row with a window seat in Premium Econ. She returned the JS Form and asked me to go to the captain and have him sign it off. Went to the cockpit introduced myself to the Capt and 2 FO's. They were more than happy to invite me for the ride. Took my seat and 8hrs later I was in NZ.

On the way back you do not have to fill out a JS form or check in with the Crew cause the gate agents in AKL won't let you go to the cockpit. So just go as a regular non-rev.
This is way cheaper than ZED or ID on HAL. Its just that you have to make the agents work because they will try to deny you even though it can be done. Just show them the ALPA JS website or if you have the Jumpseater APP on the Iphone. Tell them that you can pay at fees/taxes at the counter!
I just came back and it was so amazing. Went to Rotorua and Lake Taupo, did some bungy jumping. Awesome and cheap trip! PM if you have any other questions!!
Safe Flying!