I would make an educated guess that you would need to submit the employment contract, and additionally a new
Employer Supplementary Form signed by the new employer (you would have submitted one of these relating to your previous job when lodging your SMC application initially). The ANZSCO occupation title, occupation code and skill level is specified on the Employer Supplementary Form. I would, were I in this position, also submit a cover letter that gives a brief outline of the additional information being provided, and that includes my INZ client number and SMC EOI reference number so that INZ would be able to accurately match the additional information to the SMC application in progress.
I would suggest that you contact the INZ call centre to be sure of the above (and please post the response here!).
EDIT: Just to remind and reiterate, the Employer Supplementary Form must be completed and signed by your employer (not by you!). Your employer must then return the completed form to you -- submitting it to INZ is
your responsibility. This is all outlined on the cover page on the Employer Supplementary Form.