It featured several photos of Neve with her parents, and one with midwife Libby.
"Shoutout to my mum's midwife Libby for helping me get here safely! #day2 #ilovemylifealready," one post said.
The latest post marked the end of Neve's first day.
"Today marks my first day into the real world... I slept through most of it though."
Ardern's press secretary Andrew Campbell said the account had not been authorised by the Prime Minister nor Clarke.
The Instagram account was reminiscent of social media accounts set up for Ardern and Gayford's late cat Paddles.
Paddles, a ginger SPCA cat, had become a celebrity in its own right, and a Twitter account had been established in its name.
However, it died in November last year after being struck by a car, sparking headlines across the globe.
Ardern said yesterday, like most parents, she and Gayford went through a struggle of trying to decide on the name, but settled on Neve Te Aroha Ardern Gayford, with Ardern acting as a middle name and Gayford the surname.
"We chose Neve because we just liked it and when we met her we thought she looked like she suited the name."
Ardern said the name meant bright and radiant, as well as snow, which they thought suited the time of year - being Matariki and winter solstice.
They also chose Te Aroha in tribute to the amount of love the baby had been shown.
"All of the names we were gifted along the way, I thought how do I reflect the generosity particularly from all of the iwi who gifted us names.
"Te Aroha seemed to be a way that we could show that love and generosity. It is also the place where all my family are from and I grew up under that mountain," she said of her Waikato roots.
During her six weeks of maternity leave, Winston Peters will be filling in as acting prime minister.
When Ardern returns to work, Gayford will then step in as fulltime stay-at-home father.
