Although couples tend to think of the EOI as 'OUR application', that isn't so for INZ. One person is the applicant, and the spouse is just an et cetera who is admitted as part of the deal. So no, you can't change who is primary applicant.
The way INZ has the quotas set at the moment, anyone who has no job offer has to have an EOI with 140 points for it to be selected. With a job offer, an EOI with a minimum of 100 points will be selected. There is no point, if neither of you has a job offer, of putting in the EOI, unless you can claim 140 points.
However, either partner could apply for a visitor's visa to job-hunt. This is an entirely acceptable use for a visitor's visa, as long as people are honest about it.
http://www.immigration.govt.nz/NR/rd...20June2014.pdf
If you have the 140 points and have put in the EOI, either partner could still go if they are granted a visitor's visa in that way.
If the main applicant OR the partner gets offered a skilled job in NZ, s/he can apply for a temporary work visa, which can be processed quite quickly and allow the employment to start, and at the same time, the partner can apply for partner-sponsored temporary work visa, linking the two with a covering letter so it will all be worked as one case. This situation could mean that both parties get to be working in NZ while the original application under SMC is still ticking along in the background, taking the time that it takes.