In order to get a post-study employer-assisted work visa, the job has to be related to your NZ studies. For an Essential Skills work visa it needs to be on the ANZSCO and you need the apprpriate qualifications and/or experience as specified for that job. For residence under SMC, the job also has to be on the List of Skilled Occupations. None of them require the job to be either on the skills shortage list or with an accredited employer. Have a play with
http://skillshortages.immigration.govt.nz/ to see what might be possible. In terms of a partnership-based temporary visa, you need to have evidence that you been living together in a genuine and stable relationship for at least 3 months. If after that you got a suitable job, you could apply for residence in your own right. Alternatively, once you had been together for a year, you could apply for residence as a partner. (I have assumed your partner is a NZ resident or citizen). Hope that helps.