The second round of permits are now available on the Service NSW website.
Albury and Wodonga residents have reported being able to obtain a Border zone resident permit without any difficulty.
The new rules mean that from Wednesday, they will only be able to use the permits to cross into NSW for work, education or health reasons.
A map released on Monday showed towns such as Rutherglen, Barnawartha and Bethanga as also being eligible for a Border zone resident permit.
But when the Service NSW website went live at 4pm Tuesday, some people with addresses in those towns were not given the option to register as a Border zone resident.
NSW residents were also not give the option to obtain a critical services permit, which would allow them to return home without needing to isolate for 14 days.
More to come.
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