"on a second look" is correct and usable in written English. You can use it when you want to refer to looking at something again, usually with more thought or attention. For example, "On a second look, I decided the dress didn't suit me after all."
On first glance, one might think that Cenci's shirts looked like every other well-made shirt, but on a second look one is struck by how the colors are just a little unusual, or the cut maybe just a touch more indulgent than one is used to seeing in a Jermyn Street shirt.
On a second look, he spotted two more teeth and a shard from a fish jaw.
On a second look, he saw a man standing in the street shooting downward.
The effect is almost like something drawn by a child, but on a second look, far from artless.
But he admitted: "We now believe, based on a second look, that our team has made an error in judgment -- an honest error, but an error nonetheless.
One of the studies that, at first glance, shows aluminium is present in breast tissue is, on a second look, inconclusive because the authors didn't compare normal (non-cancerous) tissue.
Motorcycle makers Alta Motors have opened the doors to their new factory in Brisbane, California, and TechCrunch was on the scene for a first look.
When the conservative National Review polled its readers on whether he deserved a second look, the vast majority said no.
John already gave you his "quick look" on the 8520, but a second look never hurts.
Even though your partner might be utterly helpful and willing to disclose all on the surface, take a second look.
Thursday: Weekly initial jobless claims hit The Street on Thursday, followed by a second look at first-quarter gross domestic product.
Ludwig does not simply clarify my doubts with English writing, it enlightens my writing with new possibilities
Simone Ivan Conte
Software Engineer at Adobe, UK