I am not clear whether you want to pursue your postgraduate studies or undergraduate studies in New Zealand. If it is postgraduate studies, it is very difficult to get admitted in any of the reputed universities of New Zealand with 6.0 band score in IELTS. If you want to continue your undergraduate studies, then there are many universities that accept international students with overall 6.0 band score in academic IELTS.

Some the universities that accept international students with 6.0 band score in IELTS are listed below:

University of Auckland
University of Auckland accepts international undergraduate students with IELTS score of 6.0 in all its programs except in some of the pre specified programs like Engineering, Nursing, Medicine, Surgery programs.

University of Canterbury
The minimum performance levels for admission of international students in University of Canterbury is 6.0 band score in IELTS with not less than 6.0 in any individual section of academic IELTS.

University of Otago
University of Otago also accepts international undergraduate students having IELTS score of 6.0 in the academic module with no individual section band less than 6.0.

Massey University
Massey University also accepts international undergraduate students with minimum overall band score of 6.0 with no individual band score less than 5.5.

Cranfield University
Cranfield University requires overall IELTS score of 6.0 with no band less than 5.5 for its all international undergraduate students.

Victoria University of Wellington
Victoria University of Wellington also accepts all international undergraduates with IELTS band score of 6.0 except for degrees provided by faculty of law and education.