I got a job offer while I was still overseas, before my visa application was through.
Be honest, put it in your resume (legality to work in New Zealand in application or something similar) and mention it in your cover letter. And be prepared for a lot of negative replies (I have noticed that NZ employers do send "sorry" emails back). Sign up for those automatic job alert emails if you can (many bigger companies/institutions do this). Maybe contact employment agencies and recruiters if that fits. And keep a regular scan through job websites (e.g. weekly at
http://www2.careers.govt.nz/index.php?id=2059). Don't give up. If anyone is interested, a 'Skype' interview or phone interview can be starting point. I went on for months like this, until one day, I applied for the job that I am now doing. Once I had the job offer in by email, I contacted my ITA branch, less than 2 weeks later, I had the resident visa, another 2 weeks later, I was in NZ.
And if I would not have gotten the job offer? The ITA would still have been processed, but at much later date, and I still would have gotten to NZ, at a much later date... still looking for a job...